Module 17. Diffusion in solids III. Lecture 17. Diffusion in solids III

Size: px
Start display at page:

Download "Module 17. Diffusion in solids III. Lecture 17. Diffusion in solids III"

Transcription

1 Module 17 Diffusion in solids III Lecture 17 Diffusion in solids III 1 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

2 Keywords: Numerical problems in diffusion, correlation between melting point and self diffusion coefficient, estimation of inter diffusion coefficients from concentration profile, Introduction In the last two modules the mechanism of diffusion in solids has been introduced. It is a solid state mixing process where vacancy plays an important role. It is governed by a set of laws. We have looked at the nature of its solution under different initial and boundary conditions. However in order to appreciate its importance and have a deeper insight it is necessary that we try to solve a few numerical problems. In this module we therefore shall look at a few problems and their solutions. Problem 1: Melting point, crystal structure, and diffusivity in terms of D & Q (molar activation energy) are given in the following table. Estimate D at a given temperature plot D versus melting point. What do you conclude from this? FCC mp C D m 2 /sec Q kj/mole Cu E Ag E Pb E 4 19 Al E Ni E BCC mp K D m 2 /sec Q kj/mole W E V E 5 38 Cr E 5 36 Nb E 4 41 Mo E Fe E This problem illustrates the effect of melting point and crystal structure of self diffusion coefficient at any given temperature. From the previous lectures we may recall that diffusivity at a given temperature T is given by. The two tables give the values of D & Q for a few metals having FCC a few others having BCC structure. Remember that R is the universal gas constant. It is equal to 8.31J/mole/ K. Copy the values to an excel sheet and calculate diffusivity at any given temperature (say 1 C). Do not forget to convert temperature to K. The melting points of FCC metals are given in C. 2 D cm 2 /sec 1E-15 1E-3 1E-45 1E-6 1E-75 1E mp deg C FCC BCC Fig 1: Shows the effect of crystal structure & melting point of metals on its self diffusion coefficient. The diffusivity at a given temperature for all metals in the two tables has been plotted against their melting points in C. The data for a given crystal structure follows a linear trend. Higher the melting Q kj/mole FCC BCC mp deg C Fig 2: Shows the effect of crystal structure & melting point of metals on its activation energy for self diffusion. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

3 point lower is its diffusivity. Most of the metals included in table for FCC structure have relatively low melting point. Nevertheless if you extend the plot for FCC metal to higher temperature you find that diffusivity in BCC structure is higher than that in FCC structure. If you look at fig 2 you find that activation energy increases with melting point for both BCC & FCC metals. Here too the trend is linear. Note that the slopes of the two plots are nearly the same. This shows that in general activation energy for self diffusion of FCC structure is higher than that of BCC metals. We know that packing density of BCC metals is a little lower than that of FCC metals. It has more vacant space therefore thermal activation needed for diffusion is lower. Consequently diffusivity of BCC structure is higher than that of FCC structure. This also suggests if you need diffusivity data for a metal and you do not have access to diffusivity data base for all metals you could use plots as above to find diffusivity if its melting point is known. Problem 2: In diffusion experiment the markers between two metals A & B moves through a distance of.144 cm in 2 hrs with respect to the Matano interface. If inter diffusion coefficient for atom fraction A =.4 is 1 7 cm 2 /sec and the slope of the concentration profile at this point is 2. /cm estimate intrinsic diffusivities of A & B..144 cm Fig 3: A sketch to illustrate how the diffusion couple with markers looks like and the concentration of B in atom fraction as a function of distance at a time t. N B.4 Slope = 2. t = 2 hr x Marker velocity. 21/ cm 2 /sec (1) 11 cm 2 /sec (2) 1 (3) Since.6 &.4 are known; it is possible to solve equation 1 & equation & 9.41 In such problems you need to pay proper attention to the direction in which the marker moves. If it shifts along the positive direction velocity is positive. If it moves in the opposite direction it is negative. In this case D B > D A therefore more number of B atoms move towards left than that of A towards right. 3 Problem 3: The concentration profile in a diffusion experiment conducted couple of consisting of metal A & B is given in the following Fig 4 as a function of distance measured from the Matano interface after t = 5hrs. Calculate inter diffusion coefficient as a function of composition. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

4 The aim of this example is to illustrate how diffusivity can be estimated from composition versus distance plot at a given time. The first step is to find the location of the Matano interface. If there is no void formation it refers to the original interface between the two metals A & B. 1 d e composition c Fig 4: Concentration of B as a function of distance one end of the diffusion couple Slide 1 illustrates the procedure for the evaluation of diffusivity from the concentration versus distance plot. Often the composition is reported in weight percent. However it would be more appropriate to convert it into atom fraction. Let us assume the atomic weights of A & B are nearly the same. In such a case weight fraction and atom fraction are approximately the same. Problem 3 : Solution method 1 a a b c d e b.5 distance from one end Slide 1: This explains the method to be followed to estimate diffusivity. We need to find the area under the plot defined by. This is done either graphically or numerically if the values could be read from the graph. We also need to find the slope of the plot at a given value of c as shown. If these two are known diffusivity can be calculated since time t is known. The distance x is to be measured from the Matano plane. This divides the plot into two parts having the same area. A look at fig 4 suggests that it could be given by the dotted line. 4 The plot in fig 4 can be divided into two parts of equal area. In this case the area enclosed by abc is nearly same as that within cde. Therefore the dotted line may be taken to the Matano plane. With respect to this find the distances at which the compositions are given. Those who are conversant with spreadsheet can do this easily by entering the composition as a function of distance. Table 1 shows how a spreadsheet would look like. Note that c is the same for all the elemental strips whose areas are to be estimated. Here x denotes the length of the strip along x axis. Find the average values of x for each of the strips. These are entered as xav in table 1. The product gives the area of a strip. Table 1 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

5 shows areas of each of these strips separately. Note that some of these are positive and some are negative. If the location of Matano plane is correctly entered the sum total area should be zero. In this case the location of this plane is.393. The column after area in table 1 gives the total areas on the two sides of Matano plane. Since these two are equal it may be concluded that it is the correct location of the Matano plane. Figure 5 gives the concentration profile with respect to the distance from the Matano plane. Matano Plane Table 1: Spread sheet table illustrating how to estimate diffusivity at different concentrations.393 dist. x % comp c xav area x cx mod area sum area D E E E E E E E E E E E E E E E composition Distance from Matano plane 5 Fig 5: Composition of B as a function of distance from the Matano plane. We also need to find the slope as a function of concentration. The column c/x gives the slope at different values of c. This is obtained by dividing the numerical values under the respective columns. The NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

6 integral is evaluated by adding the areas of elemental strips in column mod area. The expression which is to be used to find diffusivity is as follows: (4) The area is to be evaluated from c= when x is negative. Therefore initially it is positive (note there is a negative sign in the equation 4) and it keeps increasing as it approaches Matano plane. This is where it attains its maximum value. Thereafter it should keep decreasing. The values under column sum area in table 1 show such a trend. Now that we know the time (t = 5hrs), integral (with sign), and the slope as function of composition(c) the diffusivity can be found using the equation 4. The values thus obtained are given in column D. Note that the slope at the both ends of the composition distance plot are nearly zero. Therefore the error in the estimation of diffusivity in these regions is likely to be high. This is therefore not suitable for the estimation of diffusivity at very low and very high concentration. Figure 6 gives diffusivity as a function of concentration in this system. 2.E-7 1.5E-7 D 1.E-7 5.E-8-1.6E % Composition Fig 6: Shows diffusivity D as a function of composition Radioactive tracers have been used to find self diffusion coefficients. It is customary to denote tracer diffusivity of an element with asterisk sign. In this case tracer diffusivity of A & B can be denoted as &. The relation between self diffusion coefficient and intrinsic diffusivities are given by: 6 1 (5) 1 (6) NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

7 D A & D B are intrinsic diffusion coefficients, A & B are activity coefficients and N A & N B are atom fractions. The well known Gibbs Duhem equation gives a relation between activity coefficients of the two constituents in a binary solution. This is given by: ln ln (7) In a binary alloy the relation between atom fractions is. Therefore (8) Therefore the expression for inter diffusion coefficient is given by: 1 1 (9) Or 1 (1) The inter diffusion coefficient can therefore be evaluate from self diffusion coefficient of the two components and their activity coefficient. This has been found to be valid for several systems (example Au Ni alloy where the methods give similar result till N Ni =.7). For ideal solution activity coefficient is equal to one. Therefore in such a system the following relation would be valid: (11) This shows that when N B approaches zero inter diffusion coefficient is equal to D B and at N A = ;. Diffusion is a process of mixing or alloying. If it is an ideal system where activity is equal to atom fraction the inter diffusion coefficient versus atom fraction plot should be linear. This is illustrated in slide 2. 7 Thermodynamic factor: non ideal solution D = N B D A + N A D B = N B D A +(1-N B )D B D B N B 1 d ln A D DANB DBNA1 dln NA D A Slide 2: Illustrates effect of composition on inter diffusion coefficient. In an ideal system the plot is linear. However in a non ideal system the plot is expected to deviate from linearity. Depending on the nature of deviation it may be higher or lower than the inter diffusion coefficient of an ideal system as shown in the sketch. If The problem that we solved exhibits negative deviation. In this case;. whereas if. The term 1 is the thermodynamic factor. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

8 Problem 4: Consider a 1D (one dimensional) lattice. Simulate diffusion as random walk along positive & negative directions with equal probability (=.5 each). Find the mean distance covered & its standard deviation as a function of time (number of steps). Compare the results with normal distribution. Fig 7: A sketch representing a one dimensional lattice. An atom at position can move forward or backward with equal probability We have seen that the movement of atoms in diffusion can be viewed as a random walk process. The purpose of this problem is to illustrate how to find the location of the atom after certain number of jumps along the line in fig 7 using this concept. In each jump it moves a unit distance in any of the two directions, forward or backward. Since both are equally likely the probability of moving forward or backward may be taken as.5. A jump may be assumed to represent a time step. Therefore it denotes both time and the total distance covered by the atom. However if you find its location after t jumps and measure its distance along the line you would notice that it is much less than t. We would try to estimate this. Such problems can easily be simulated using electronic spread sheet such as excel. It has a random number generator. This can be used to decide whether the atom from its present location would move forward or backward. Repeat the process for a large number of times and find its final location. Each movement of atom is one step. Let the process be repeated for a set of n atoms considering their movements to be independent of each other. Slide 3 illustrates how this could be setup in excel. 8 Slide 3: Describes how to setup an electronic Construction of matrix X(t,n) in Excel X(t,n) = location of atom at time t in nth trial spreadsheet to solve such a problem. It has several cells arranged in rows and columns. A B C.. Enter integers 1, 2, 3 till N in the first row. N columns represent the state of each of the N N atoms in N sites in a 1D lattice. The second 2 row denotes initial positions of N atoms in respective lattice. They are all zero. This indicates that these are at their initial location. The third row gives the locations of N atoms after the first time step. The values in t respective cells are decided using random number generator. Each column of the spreadsheet simulates the locations of respective atoms after every time step till t. Except for the first two rows all the cells are filled up by integers representing the locations of the atoms. The second row denotes initial locations of N atoms in respective lattice. The X(t, n) in general represents the location of nth atom after time step t and X(2, 1) denotes initial position of the first atom at t=. In order to find X(3, 1), the location of the first atom in time step 1 we need to find a random NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

9 number. This is done in excel by RAND() a standard in built function. It is a pseudo random number generator. It generates a number and 1. If this is greater than.5 the atom would move forward (from its initial position to 1). If it is less than.5 the atom moves backward (from to 1). This is done by entering the following statement in cell A2 of the excel sheet. <cell A3> = <cell A2> + IF(RAND() >.5, 1, 1) (12) Before you do this make sure to set the excel sheet calculation mode to manual. Once this is done enter the statement in equation 12 in cell A3 and copy the same to all the cells that form a part of the matrix X(t,n). Such a statement generates an integer denoting the location of the atom at a given time step by adding either +1 or 1 to the contents of the cell in the same column but in the previous row. The decision is based on the outcome of the IF statement involving RAND() in equation 12. Enter excel function AVERAGE() in column marked mean () to calculate the average of all the values from column 1 to N in the same row. Enter excel function STDV() in the column marked to calculate standard deviation of the same data set in respective rows. Now press F9 to perform calculation. This is done row wise. The column now gives the most likely location of the atoms with respect to their initial positions. The column gives the corresponding standard deviation (see Table 2). Table 2: A typical output of an excel spread sheet Table 2 gives a typical output of the spreadsheet calculation. Note that with increasing time step the mean does not change significantly. However standard deviation keeps increasing. You may try to perform this exercise for a much larger matrix to find the trend. Figure 7 gives a typical plot of mean as function of time. 9 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

10 mean time Fig 7: Shows the mean position of an atom as a function of time. Mean denotes average of 25 trials stdv time Fig 8: Shows standard deviation of the mean positions of atoms as a function of time steps time Fig 9: Shows that the square of standard deviation (variance) of the mean positions of atoms varies linearly with time. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

11 The mean denotes average or the most probable location of the atom. This has been obtained on the basis of 25 trials. The plot in fig 7 keeps changing every time you press F9 button. It is not unexpected since the process is one of random walk. Note that the mean location after around 1 time steps varies within approximately ±1.5. This represents that in spite of such a large number of jumps the average displacement is so less. As against this the trend shown by standard deviation is extremely stable. Figure 8 gives a typical plot of standard deviation a function of time. The plot does not change significantly irrespective of the number of times you repeat the calculation by pressing F9 button. The trend is parabolic. This is why the plot 2 versus time is linear (fig 9). Nevertheless on the basis of 25 trials it may be concluded the final location after nearly 1 time steps is given by a distribution having a mean and a standard deviation. The question that comes up is what is the nature of this distribution? Excel has a function called frequency which has been used to establish this. Create a column named bin. It has numbers ranging from 9 to +9. This represents class interval. Use the frequency function of excel to count the number of times the final location after 1 time steps lies between each of these. Table 3 Typical output from excel sheet having all the data on the simulation of the 1D random walk problem. Column RAND gives predicted result. Column Norm gives normal distribution plot. Bin Rand Cum Freq Norm Mean.592 stdv NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

12 Frequency Location, x Rand Norm Fig 1: A comparison of the frequency distribution obtained as a result of 1D simulation (RAND) and the corresponding normal distribution plot (Norm) obtained on the basis of the mean position and standard deviation after 1 time step. Frequency is a standard array function in excel. It looks up an array having all the data. In this it refers to the row corresponding to say 1th time step (A11:IP11). Subsequently it divides the same into the specified class interval stored the column named BIN. Here let this be A16:A115. The statement to be entered in the column marked RAND is = {= (Frequency (A11:IP11, A16:A115)}. Note that in order to enter an array function you need to press Ctrl, Shift & Enter together. Use normal distribution function to convert the mean & standard deviation to frequency distribution as shown in Table 3 under column marked NORM. Figure 1 compares the predictions based on random walk process and those calculated using normal distribution function. The standard deviation is proportional to the square root of time. The effective depth of penetration due to diffusion is also proportional to the square root of time. In short and. The constant of proportionality is related to the diffusivity. Summary 12 In the last three modules an attempt has been made to give an elementary concept of the solid state mixing process known as diffusion. This is governed by a set of laws expressed in the form of a differential equation. We have looked a 1D solution only. However the concept can easily be extended to 3D. In this lecture we have solved a few numerical problems in detail to give a deeper insight into this process. Diffusion is associated with the random movement of atoms. It has a direct correlation with several material parameters like crystal structure, melting point & enthalpy. A first estimate of diffusivity can be obtained from such data. We have also seen that diffusivity in an alloy is a function of composition. It is related to the movement of all the species present in an alloy. We however looked at binary systems only. We have solved a problem to find the diffusivities of two species where the NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

13 markers move with respect to the initial interface of the diffusion couple called the Matano plane. Finally we have tried to simulate the process of diffusion as a random walk problem. It has been shown that the standard deviation truly represents effective depth of penetration in a diffusion couple. 13 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering

Module 16. Diffusion in solids II. Lecture 16. Diffusion in solids II

Module 16. Diffusion in solids II. Lecture 16. Diffusion in solids II Module 16 Diffusion in solids II Lecture 16 Diffusion in solids II 1 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering Keywords: Micro mechanisms of diffusion,

More information

Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at

Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at Chapter 7 What is steady state diffusion? Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at any point, x, and hence the concentration gradient at x, in the solid,

More information

Stochastic Modelling

Stochastic Modelling Stochastic Modelling Simulating Random Walks and Markov Chains This lab sheets is available for downloading from www.staff.city.ac.uk/r.j.gerrard/courses/dam/, as is the spreadsheet mentioned in section

More information

Chemical Bonding Ionic Bonding. Unit 1 Chapter 2

Chemical Bonding Ionic Bonding. Unit 1 Chapter 2 Chemical Bonding Ionic Bonding Unit 1 Chapter 2 Valence Electrons The electrons responsible for the chemical properties of atoms are those in the outer energy level. Valence electrons - The s and p electrons

More information

Experiment 7: Understanding Crystal Structures

Experiment 7: Understanding Crystal Structures Experiment 7: Understanding Crystal Structures To do well in this laboratory experiment you need to be familiar with the concepts of lattice, crystal structure, unit cell, coordination number, the different

More information

Entropy of bcc L, fcc L, and fcc bcc Phase Transitions of Elemental Substances as Function of Transition Temperature

Entropy of bcc L, fcc L, and fcc bcc Phase Transitions of Elemental Substances as Function of Transition Temperature Doklady Physics, Vol. 45, No. 7, 2, pp. 311 316. ranslated from Doklady Akademii Nauk, Vol. 373, No. 3, 2, pp. 323 328. Original Russian ext Copyright 2 by Udovskiœ. PHYSICS Entropy of bcc, fcc, and fcc

More information

COURSE 3.20: THERMODYNAMICS OF MATERIALS. FINAL EXAM, Dec 18, 2000

COURSE 3.20: THERMODYNAMICS OF MATERIALS. FINAL EXAM, Dec 18, 2000 NAME: COURSE 3.20: THERMODYNAMICS OF MATERIALS FINAL EXAM, Dec 18, 2000 PROBLEM 1 (10 POINTS) PROBLEM 2 (15 POINTS) PROBLEM 3 (12 POINTS) PROBLEM 4 (12 POINTS) PROBLEM 5 (12 POINTS) PROBLEM 6 (15 POINTS)

More information

Lecture 1: Atomic Diffusion

Lecture 1: Atomic Diffusion Part IB Materials Science & Metallurgy H. K. D. H. Bhadeshia Course A, Metals and Alloys Lecture 1: Atomic Diffusion Mass transport in a gas or liquid generally involves the flow of fluid (e.g. convection

More information

Lab 1: Handout GULP: an Empirical energy code

Lab 1: Handout GULP: an Empirical energy code Lab 1: Handout GULP: an Empirical energy code We will be using the GULP code as our energy code. GULP is a program for performing a variety of types of simulations on 3D periodic solids, gas phase clusters,

More information

MATSCI 204 Thermodynamics and Phase Equilibria Winter Chapter #4 Practice problems

MATSCI 204 Thermodynamics and Phase Equilibria Winter Chapter #4 Practice problems MATSCI 204 Thermodynamics and Phase Equilibria Winter 2013 Chapter #4 Practice problems Problem: 1-Show that for any extensive property Ω of a binary system A-B: d ( "# ) "# B, = "# + 1$ x B 2- If "# has

More information

Concept of the chemical potential and the activity of elements

Concept of the chemical potential and the activity of elements Concept of the chemical potential and the activity of elements Gibb s free energy, G is function of temperature, T, pressure, P and amount of elements, n, n dg G G (T, P, n, n ) t particular temperature

More information

Driving force for diffusion and Fick s laws of diffusion

Driving force for diffusion and Fick s laws of diffusion Driving force for diffusion and Fick s laws of diffusion In the last lecture, it is eplained that diffusion of elements between different blocks is possible because of difference in chemical potential

More information

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 9a BONDING AND SOLUTIONS

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 9a BONDING AND SOLUTIONS 3.091 Introduction to Solid State Chemistry Lecture Notes No. 9a BONDING AND SOLUTIONS 1. INTRODUCTION Condensed phases, whether liquid or solid, may form solutions. Everyone is familiar with liquid solutions.

More information

MECH 6661 lecture 9/1 Dr. M. Medraj Mech. Eng. Dept. - Concordia University

MECH 6661 lecture 9/1 Dr. M. Medraj Mech. Eng. Dept. - Concordia University Thermodynamic Models Multicomponent Systems Outline Thermodynamic Models Regular Solution Models Sublattice Model Associated Solutions Cluster Variation Model Quasichemical Model Cluster Expansion Model

More information

Basic chromatographic parameters and optimization in LC

Basic chromatographic parameters and optimization in LC AM0925 Assignment Basic chromatographic parameters and optimization in LC Introduction This is a computer exercise where you will apply a simulator of reversed phase LC to study the influence of chromatographic

More information

Physics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces)

Physics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces) Physics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces) Dr. Anurag Srivastava Atal Bihari Vajpayee Indian Institute of Information Technology and Manegement,

More information

Ram Seshadri MRL 2031, x6129, These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings

Ram Seshadri MRL 2031, x6129, These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings Crystals, packings etc. Ram Seshadri MRL 2031, x6129, seshadri@mrl.ucsb.edu These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings The unit cell and its propagation Materials usually

More information

Falling Bodies (last

Falling Bodies (last Dr. Larry Bortner Purpose Falling Bodies (last edited ) To investigate the motion of a body under constant acceleration, specifically the motion of a mass falling freely to Earth. To verify the parabolic

More information

Thermodynamics (Classical) for Biological Systems. Prof. G. K. Suraishkumar. Department of Biotechnology. Indian Institute of Technology Madras

Thermodynamics (Classical) for Biological Systems. Prof. G. K. Suraishkumar. Department of Biotechnology. Indian Institute of Technology Madras Thermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras Module No. #04 Thermodynamics of solutions Lecture No. #22 Partial

More information

PHYS 1111L - Introductory Physics Laboratory I

PHYS 1111L - Introductory Physics Laboratory I PHYS 1111L - Introductory Physics Laboratory I Laboratory Advanced Sheet Acceleration Due to Gravity 1. Objectives. The objectives of this laboratory are a. To measure the local value of the acceleration

More information

University of Ljubljana. Faculty of Mathematics and Physics. Department of Physics. High-entropy alloys. Author: Darja Gačnik

University of Ljubljana. Faculty of Mathematics and Physics. Department of Physics. High-entropy alloys. Author: Darja Gačnik University of Ljubljana Faculty of Mathematics and Physics Department of Physics Seminar I 1 st year of Master study programme High-entropy alloys Author: Darja Gačnik Mentor: prof. Janez Dolinšek Ljubljana,

More information

Week 11/Th: Lecture Units 28 & 29

Week 11/Th: Lecture Units 28 & 29 Week 11/Th: Lecture Units 28 & 29 Unit 27: Real Gases Unit 28: Intermolecular forces -- types of forces between molecules -- examples Unit 29: Crystal Structure -- lattice types -- unit cells -- simple

More information

MECH6661 lectures 6/1 Dr. M. Medraj Mech. Eng. Dept. - Concordia University. Fe3C (cementite)

MECH6661 lectures 6/1 Dr. M. Medraj Mech. Eng. Dept. - Concordia University. Fe3C (cementite) Outline Solid solution Gibbs free energy of binary solutions Ideal solution Chemical potential of an ideal solution Regular solutions Activity of a component Real solutions Equilibrium in heterogeneous

More information

Statistics of Radioactive Decay

Statistics of Radioactive Decay Statistics of Radioactive Decay Introduction The purpose of this experiment is to analyze a set of data that contains natural variability from sample to sample, but for which the probability distribution

More information

Lab 1: Handout GULP: an Empirical energy code

Lab 1: Handout GULP: an Empirical energy code 3.320/SMA 5.107/ Atomistic Modeling of Materials Spring 2003 1 Lab 1: Handout GULP: an Empirical energy code We will be using the GULP code as our energy code. GULP is a program for performing a variety

More information

PHYS 2211L - Principles of Physics Laboratory I

PHYS 2211L - Principles of Physics Laboratory I PHYS 2211L - Principles of Physics Laboratory I Laboratory Advanced Sheet Acceleration Due to Gravity 1. Objectives. The objectives of this laboratory are a. To measure the local value of the acceleration

More information

Lecture 09 Combined Effect of Strain, Strain Rate and Temperature

Lecture 09 Combined Effect of Strain, Strain Rate and Temperature Fundamentals of Materials Processing (Part- II) Prof. Shashank Shekhar and Prof. Anshu Gaur Department of Materials Science and Engineering Indian Institute of Technology, Kanpur Lecture 09 Combined Effect

More information

1. In Activity 1-1, part 3, how do you think graph a will differ from graph b? 3. Draw your graph for Prediction 2-1 below:

1. In Activity 1-1, part 3, how do you think graph a will differ from graph b? 3. Draw your graph for Prediction 2-1 below: PRE-LAB PREPARATION SHEET FOR LAB 1: INTRODUCTION TO MOTION (Due at the beginning of Lab 1) Directions: Read over Lab 1 and then answer the following questions about the procedures. 1. In Activity 1-1,

More information

Determination of Density 1

Determination of Density 1 Introduction Determination of Density 1 Authors: B. D. Lamp, D. L. McCurdy, V. M. Pultz and J. M. McCormick* Last Update: February 1, 2013 Not so long ago a statistical data analysis of any data set larger

More information

Experiment 3. d s = 3-2 t ANALYSIS OF ONE DIMENSIONAL MOTION

Experiment 3. d s = 3-2 t ANALYSIS OF ONE DIMENSIONAL MOTION Experiment 3 ANALYSIS OF ONE DIMENSIONAL MOTION Objectives 1. To establish a mathematical relationship between the position and the velocity of an object in motion. 2. To define the velocity as the change

More information

OLI Tips #52 Alloy Manager

OLI Tips #52 Alloy Manager The Right Chemistry OLI Tips #52 Alloy Manager Calculation of the activity of individual components in alloys. The development of this activity model was performed at the Oa Ridge National Laboratory.

More information

Name(s): Date: Course/Section: Mass of the Earth

Name(s): Date: Course/Section: Mass of the Earth Name(s): Date: Course/Section: Grade: Part 1: The Angular Size of the Earth Mass of the Earth Examine the image on the lab website. The image of the Earth was taken from the Moon on Aug 23, 1966 by Lunar

More information

LAB 2 - ONE DIMENSIONAL MOTION

LAB 2 - ONE DIMENSIONAL MOTION Name Date Partners L02-1 LAB 2 - ONE DIMENSIONAL MOTION OBJECTIVES Slow and steady wins the race. Aesop s fable: The Hare and the Tortoise To learn how to use a motion detector and gain more familiarity

More information

Linear Motion with Constant Acceleration

Linear Motion with Constant Acceleration Linear Motion 1 Linear Motion with Constant Acceleration Overview: First you will attempt to walk backward with a constant acceleration, monitoring your motion with the ultrasonic motion detector. Then

More information

Cryogenic Engineering Prof. M. D. Atrey Department of Mechanical Engineering Indian Institute of Technology, Bombay. Lecture No. #23 Gas Separation

Cryogenic Engineering Prof. M. D. Atrey Department of Mechanical Engineering Indian Institute of Technology, Bombay. Lecture No. #23 Gas Separation Cryogenic Engineering Prof. M. D. Atrey Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. #23 Gas Separation So, welcome to the 23rd lecture, on Cryogenic Engineering,

More information

Introduction to Determining Power Law Relationships

Introduction to Determining Power Law Relationships 1 Goal Introduction to Determining Power Law Relationships Content Discussion and Activities PHYS 104L The goal of this week s activities is to expand on a foundational understanding and comfort in modeling

More information

Inorganic Exam 1 Chm October 2010

Inorganic Exam 1 Chm October 2010 Inorganic Exam 1 Chm 451 28 October 2010 Name: Instructions. Always show your work where required for full credit. 1. In the molecule CO 2, the first step in the construction of the MO diagram was to consider

More information

Determining the Rate Law and Activation Energy for the Methyl Blue Reaction:

Determining the Rate Law and Activation Energy for the Methyl Blue Reaction: Experiment 4 Determining the Rate Law and Activation Energy for the Methyl Blue Reaction: Pre-lab Assignment Before coming to lab: Read the lab thoroughly. An exercise in experimental design Answer the

More information

Regression Analysis and Forecasting Prof. Shalabh Department of Mathematics and Statistics Indian Institute of Technology-Kanpur

Regression Analysis and Forecasting Prof. Shalabh Department of Mathematics and Statistics Indian Institute of Technology-Kanpur Regression Analysis and Forecasting Prof. Shalabh Department of Mathematics and Statistics Indian Institute of Technology-Kanpur Lecture 10 Software Implementation in Simple Linear Regression Model using

More information

Thermodynamic parameters in a binary system

Thermodynamic parameters in a binary system Thermodynamic parameters in a binary system Previously, we considered one element only. Now we consider interaction between two elements. This is not straightforward since elements can interact differently

More information

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 6a BONDING AND SURFACES

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 6a BONDING AND SURFACES 3.091 Introduction to Solid State Chemistry Lecture Notes No. 6a BONDING AND SURFACES 1. INTRODUCTION Surfaces have increasing importance in technology today. Surfaces become more important as the size

More information

Experimental design (DOE) - Design

Experimental design (DOE) - Design Experimental design (DOE) - Design Menu: QCExpert Experimental Design Design Full Factorial Fract Factorial This module designs a two-level multifactorial orthogonal plan 2 n k and perform its analysis.

More information

Static and Kinetic Friction

Static and Kinetic Friction Experiment Static and Kinetic Friction Prelab Questions 1. Examine the Force vs. time graph and the Position vs. time graph below. The horizontal time scales are the same. In Region I, explain how an object

More information

Switch + R. ower upply. Voltmete. Capacitor. Goals. Introduction

Switch + R. ower upply. Voltmete. Capacitor. Goals. Introduction Lab 6. Switch RC Circuits ower upply Goals To appreciate the capacitor as a charge storage device. To measure the voltage across a capacitor as it discharges through a resistor, and to compare + the result

More information

Computer simulation of radioactive decay

Computer simulation of radioactive decay Computer simulation of radioactive decay y now you should have worked your way through the introduction to Maple, as well as the introduction to data analysis using Excel Now we will explore radioactive

More information

(Refer Slide Time: 00:00:43 min) Welcome back in the last few lectures we discussed compression refrigeration systems.

(Refer Slide Time: 00:00:43 min) Welcome back in the last few lectures we discussed compression refrigeration systems. Refrigeration and Air Conditioning Prof. M. Ramgopal Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture No. # 14 Vapour Absorption Refrigeration Systems (Refer Slide

More information

Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras

Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras Module No. # 07 Lecture No. # 04 Ordinary Differential Equations (Initial Value

More information

Competing Models, the Crater Lab

Competing Models, the Crater Lab Competing Models, the Crater Lab Graphs, Linear Regression, and Competing Models PHYS 104L 1 Goal The goal of this week s lab is to check your ability to collect data and use graphical analysis to determine

More information

PHY221 Lab 2 - Experiencing Acceleration: Motion with constant acceleration; Logger Pro fits to displacement-time graphs

PHY221 Lab 2 - Experiencing Acceleration: Motion with constant acceleration; Logger Pro fits to displacement-time graphs Page 1 PHY221 Lab 2 - Experiencing Acceleration: Motion with constant acceleration; Logger Pro fits to displacement-time graphs Print Your Name Print Your Partners' Names You will return this handout to

More information

Bravais Lattice + Basis = Crystal Structure

Bravais Lattice + Basis = Crystal Structure Bravais Lattice + Basis = Crystal Structure A crystal structure is obtained when identical copies of a basis are located at all of the points of a Bravais lattice. Consider the structure of Cr, a I-cubic

More information

Lab 6. RC Circuits. Switch R 5 V. ower upply. Voltmete. Capacitor. Goals. Introduction

Lab 6. RC Circuits. Switch R 5 V. ower upply. Voltmete. Capacitor. Goals. Introduction Switch ower upply Lab 6. RC Circuits + + R 5 V Goals Capacitor V To appreciate the capacitor as a charge storage device. To measure the voltage across a capacitor as it discharges through a resistor, and

More information

CHM112 Lab Iodine Clock Reaction Part 2 Grading Rubric

CHM112 Lab Iodine Clock Reaction Part 2 Grading Rubric Name Team Name CHM112 Lab Iodine Clock Reaction Part 2 Grading Rubric Criteria Points possible Points earned Lab Performance Printed lab handout and rubric was brought to lab 3 Initial concentrations completed

More information

Chemical Reaction Engineering Prof. JayantModak Department of Chemical Engineering Indian Institute of Science, Bangalore

Chemical Reaction Engineering Prof. JayantModak Department of Chemical Engineering Indian Institute of Science, Bangalore Chemical Reaction Engineering Prof. JayantModak Department of Chemical Engineering Indian Institute of Science, Bangalore Module No. #05 Lecture No. #29 Non Isothermal Reactor Operation Let us continue

More information

2: SIMPLE HARMONIC MOTION

2: SIMPLE HARMONIC MOTION 2: SIMPLE HARMONIC MOTION Motion of a mass hanging from a spring If you hang a mass from a spring, stretch it slightly, and let go, the mass will go up and down over and over again. That is, you will get

More information

Exponential Functions

Exponential Functions CONDENSED LESSON 5.1 Exponential Functions In this lesson, you Write a recursive formula to model radioactive decay Find an exponential function that passes through the points of a geometric sequence Learn

More information

Using Microsoft Excel

Using Microsoft Excel Using Microsoft Excel Objective: Students will gain familiarity with using Excel to record data, display data properly, use built-in formulae to do calculations, and plot and fit data with linear functions.

More information

Dry mix design. Lecture Notes in Transportation Systems Engineering. Prof. Tom V. Mathew. 1 Overview 1. 2 Selection of aggregates 1

Dry mix design. Lecture Notes in Transportation Systems Engineering. Prof. Tom V. Mathew. 1 Overview 1. 2 Selection of aggregates 1 ry mix design Lecture Notes in Transportation Systems Engineering Prof. Tom V. Mathew Contents 1 Overview 1 2 Selection of aggregates 1 3 Aggregate gradation 2 4 Proportioning of aggregates 2 5 Example

More information

Classification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava

Classification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava Classification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus

More information

After successfully completing this laboratory assignment, including the assigned reading, the lab

After successfully completing this laboratory assignment, including the assigned reading, the lab University of California at Santa Cruz Jack Baskin School of Engineering Electrical Engineering Department EE-145L: Properties of Materials Laboratory Lab 6: Temperature Dependence of Semiconductor Conductivity

More information

Experiment IV. To find the velocity of waves on a string by measuring the wavelength and frequency of standing waves.

Experiment IV. To find the velocity of waves on a string by measuring the wavelength and frequency of standing waves. Experiment IV The Vibrating String I. Purpose: To find the velocity of waves on a string by measuring the wavelength and frequency of standing waves. II. References: Serway and Jewett, 6th Ed., Vol., Chap.

More information

The Coefficient of Friction

The Coefficient of Friction The Coefficient of Friction OBJECTIVE To determine the coefficient of static friction between two pieces of wood. To determine the coefficient of kinetic friction between two pieces of wood. To investigate

More information

ES205 Analysis and Design of Engineering Systems: Lab 1: An Introductory Tutorial: Getting Started with SIMULINK

ES205 Analysis and Design of Engineering Systems: Lab 1: An Introductory Tutorial: Getting Started with SIMULINK ES205 Analysis and Design of Engineering Systems: Lab 1: An Introductory Tutorial: Getting Started with SIMULINK What is SIMULINK? SIMULINK is a software package for modeling, simulating, and analyzing

More information

Electric Fields and Equipotentials

Electric Fields and Equipotentials OBJECTIVE Electric Fields and Equipotentials To study and describe the two-dimensional electric field. To map the location of the equipotential surfaces around charged electrodes. To study the relationship

More information

Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur

Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Lecture 51 Plackett Burman Designs Hello, welcome. We

More information

Introductory College Chemistry

Introductory College Chemistry Introductory College Chemistry This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to

More information

SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 215

SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 215 SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 5 Updated May 0 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of

More information

sec x dx = ln sec x + tan x csc x dx = ln csc x cot x

sec x dx = ln sec x + tan x csc x dx = ln csc x cot x Name: Instructions: The exam will have eight problems. Make sure that your reasoning and your final answers are clear. Include labels and units when appropriate. No notes, books, or calculators are permitted

More information

Lab #15: Introduction to Computer Aided Design

Lab #15: Introduction to Computer Aided Design Lab #15: Introduction to Computer Aided Design Revision: 02 Nov 2016 Print Name: Section: GETTING FAMILIAR WITH YOUR BASYS3 DIGILAB BOARD. Problem 1: (26 points) Visually inspect the Digilab board, enter

More information

CHAPTER 2. NATURE OF MATERIALS

CHAPTER 2. NATURE OF MATERIALS Materials for Civil and Construction Engineers 4th Edition Mamlouk SOLUTIONS MANUAL Full clear download (no formatting errors) at: https://testbankreal.com/download/materials-for-civiland-construction-engineers-4th-edition-mamlouksolutions-manual/

More information

INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR. NPTEL National Programme on Technology Enhanced Learning. Probability Methods in Civil Engineering

INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR. NPTEL National Programme on Technology Enhanced Learning. Probability Methods in Civil Engineering INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR NPTEL National Programme on Technology Enhanced Learning Probability Methods in Civil Engineering Prof. Rajib Maity Department of Civil Engineering IIT Kharagpur

More information

10, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics)

10, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Subect Chemistry Paper No and Title Module No and Title Module Tag 0, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) 0, Free energy

More information

Lecture 4: Classical Illustrations of Macroscopic Thermal Effects

Lecture 4: Classical Illustrations of Macroscopic Thermal Effects Lecture 4: Classical Illustrations of Macroscopic Thermal Effects Heat capacity of solids & liquids Thermal conductivity Irreversibility References for this Lecture: Elements Ch 3,4A-C Reference for Lecture

More information

Thermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras

Thermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras Thermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras Module No. # 04 Thermodynamics of Solutions Lecture No. # 25

More information

Diffusion. Diffusion. Diffusion in Solid Materials

Diffusion. Diffusion. Diffusion in Solid Materials Atoms movements in materials Diffusion Movement of atoms in solids, liquids and gases is very important Eamples: Hardening steel, chrome-plating, gas reactions, Si wafers.. etc. We will study: Atomic mechanisms

More information

Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we

More information

COUNTING ERRORS AND STATISTICS RCT STUDY GUIDE Identify the five general types of radiation measurement errors.

COUNTING ERRORS AND STATISTICS RCT STUDY GUIDE Identify the five general types of radiation measurement errors. LEARNING OBJECTIVES: 2.03.01 Identify the five general types of radiation measurement errors. 2.03.02 Describe the effect of each source of error on radiation measurements. 2.03.03 State the two purposes

More information

PHYS 275 Experiment 2 Of Dice and Distributions

PHYS 275 Experiment 2 Of Dice and Distributions PHYS 275 Experiment 2 Of Dice and Distributions Experiment Summary Today we will study the distribution of dice rolling results Two types of measurement, not to be confused: frequency with which we obtain

More information

Introduction. Pre-Lab Questions: Physics 1CL PERIODIC MOTION - PART II Spring 2009

Introduction. Pre-Lab Questions: Physics 1CL PERIODIC MOTION - PART II Spring 2009 Introduction This is the second of two labs on simple harmonic motion (SHM). In the first lab you studied elastic forces and elastic energy, and you measured the net force on a pendulum bob held at an

More information

MatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing

MatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing MatSci 331 Homework 4 Molecular Dynamics and Monte Carlo: Stress, heat capacity, quantum nuclear effects, and simulated annealing Due Thursday Feb. 21 at 5pm in Durand 110. Evan Reed In this homework,

More information

General Chemistry (Second Quarter)

General Chemistry (Second Quarter) General Chemistry (Second Quarter) This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence

More information

Diffusion in Dilute Alloys

Diffusion in Dilute Alloys Chapter 3 Diffusion in Dilute Alloys Our discussion of the atomistic mechanisms for diffusion has been confined to the situations where the diffusing species is chemically identical to the host atom. We

More information

Introduction to Physics Physics 114 Eyres

Introduction to Physics Physics 114 Eyres What is Physics? Introduction to Physics Collecting and analyzing experimental data Making explanations and experimentally testing them Creating different representations of physical processes Finding

More information

CHAPTER 2: ENERGY BANDS & CARRIER CONCENTRATION IN THERMAL EQUILIBRIUM. M.N.A. Halif & S.N. Sabki

CHAPTER 2: ENERGY BANDS & CARRIER CONCENTRATION IN THERMAL EQUILIBRIUM. M.N.A. Halif & S.N. Sabki CHAPTER 2: ENERGY BANDS & CARRIER CONCENTRATION IN THERMAL EQUILIBRIUM OUTLINE 2.1 INTRODUCTION: 2.1.1 Semiconductor Materials 2.1.2 Basic Crystal Structure 2.1.3 Basic Crystal Growth technique 2.1.4 Valence

More information

NAGRAJ SHESHGIRI KULKARNI

NAGRAJ SHESHGIRI KULKARNI INTRINSIC DIFFUSION SIMULATION FOR SINGLE-PHASE MULTICOMPONENT SYSTEMS AND ITS APPLICATION FOR THE ANALYSIS OF THE DARKEN- MANNING AND JUMP FREQUENCY FORMALISMS By NAGRAJ SHESHGIRI KULKARNI A DISSERTATION

More information

Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture No. # 02 Conservation of Mass and Momentum: Continuity and

More information

THE CRYSTAL BALL FORECAST CHART

THE CRYSTAL BALL FORECAST CHART One-Minute Spotlight THE CRYSTAL BALL FORECAST CHART Once you have run a simulation with Oracle s Crystal Ball, you can view several charts to help you visualize, understand, and communicate the simulation

More information

Kinetics of Crystal Violet Bleaching

Kinetics of Crystal Violet Bleaching Kinetics of Crystal Violet Bleaching Authors: V. C. Dew and J. M. McCormick* From Update March 12, 2013 with revisions Nov. 29, 2016 Introduction Chemists are always interested in whether a chemical reaction

More information

3. APLICATION OF GIBBS-DUHEM EQUATION

3. APLICATION OF GIBBS-DUHEM EQUATION 3. APLICATION OF GIBBS-DUHEM EQUATION Gibbs-Duhem Equation When extensive property of a solution is given by; Q' = Q'(T,P,n,n...) The change in extensive property with composition was; dq Q n dn Q n dn

More information

The OptiSage module. Use the OptiSage module for the assessment of Gibbs energy data. Table of contents

The OptiSage module. Use the OptiSage module for the assessment of Gibbs energy data. Table of contents The module Use the module for the assessment of Gibbs energy data. Various types of experimental data can be utilized in order to generate optimized parameters for the Gibbs energies of stoichiometric

More information

Mass Transfer Operations I Prof. Bishnupada Mandal Department of Chemical Engineering Indian Institute of Technology, Guwahati

Mass Transfer Operations I Prof. Bishnupada Mandal Department of Chemical Engineering Indian Institute of Technology, Guwahati Mass Transfer Operations I Prof. Bishnupada Mandal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module - 4 Absorption Lecture - 3 Packed Tower Design Part 2 (Refer Slide

More information

1. Introductory Examples

1. Introductory Examples 1. Introductory Examples We introduce the concept of the deterministic and stochastic simulation methods. Two problems are provided to explain the methods: the percolation problem, providing an example

More information

Thermodynamics of Solutions Partial Molar Properties

Thermodynamics of Solutions Partial Molar Properties MME3: Lecture 6 Thermodynamics of Solutions Partial Molar Properties A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka omposition of solutions Partial molar properties Introduction Materials

More information

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34 Linear Algebra /34 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct axes)

More information

5-Sep-15 PHYS101-2 GRAPHING

5-Sep-15 PHYS101-2 GRAPHING GRAPHING Objectives 1- To plot and analyze a graph manually and using Microsoft Excel. 2- To find constants from a nonlinear relation. Exercise 1 - Using Excel to plot a graph Suppose you have measured

More information

DIFFUSION IN SOLIDS. IE-114 Materials Science and General Chemistry Lecture-5

DIFFUSION IN SOLIDS. IE-114 Materials Science and General Chemistry Lecture-5 DIFFUSION IN SOLIDS IE-114 Materials Science and General Chemistry Lecture-5 Diffusion The mechanism by which matter is transported through matter. It is related to internal atomic movement. Atomic movement;

More information

A Scientific Model for Free Fall.

A Scientific Model for Free Fall. A Scientific Model for Free Fall. I. Overview. This lab explores the framework of the scientific method. The phenomenon studied is the free fall of an object released from rest at a height H from the ground.

More information

Figure 1. Distillation Train. Table 1. Stream compositions.

Figure 1. Distillation Train. Table 1. Stream compositions. CM 3450 Drills 2 9/8/2010 1. A stream containing compounds,, and are fed to a series of distillation columns as shown in Figure 1 with corresponding stream compositions given in Table 1. Figure 1. Distillation

More information

Web sites and Worksheets

Web sites and Worksheets Introduction Web sites and Worksheets Sandra Woodward Oakhill College Castle Hill (swoodward@oakhill.nsw.edu.au) Astronomy is a practical subject by nature. Unfortunately for teachers, many of the practical

More information

Lectures 16: Phase Transitions

Lectures 16: Phase Transitions Lectures 16: Phase Transitions Continuous Phase transitions Aims: Mean-field theory: Order parameter. Order-disorder transitions. Examples: β-brass (CuZn), Ferromagnetic transition in zero field. Universality.

More information

Newton s Second Law of Motion

Newton s Second Law of Motion Newton s Second Law of Motion Overview The purpose of this investigation is to validate Newton s Second Law of Motion. In Part A a lab cart will be accelerated by various net forces while keeping mass

More information