Thermodynamics 1 Lecture Note 1

Thermodynamics 1 Lecture Note 1 March 02, 2015 Kwang Kim Yonsei University kbkim@yonsei.ac.kr 39 8 7 34 53 Y O N Se I 88.91 16.00 14.01 78.96 126.9

Scientific Thinking - System under investigation - Description or Behavior of a system - Variables of a system - Correlation between behavior and variables of a system - Modeling of a system - Comparison of a model with a system - Revision of a model

Three NO S in class 1) No Sleep 2) No Silence, Active Class Participation 3) No Smile when asked questions

Material Science and Engineering (MSE) Final materials must perform a given task in an economical and societally acceptable manner Synthesis/ Processing Performance Composition/ Structure result of synthesis and processing Properties and performance : related to composition and structure Properties Properties governed by composition and structure

Scientific Thinking - System under investigation Gas in a balloon - Description or Behavior of a system

Scientific Thinking - System under investigation Which gas? Amount of a gas? Gas in a balloon Pressure? Volume? Temperature? Density? - Description or Behavior of a system

Chapter 1 : Properties of gases - Ideal gas : PV = nrt - Real gas : molecular interaction - Non-ideality of real gas - Deviation of real gas from ideal gas - Quantifying non-ideality - Van der Walls equation describing a non-ideal gas - Check how well Van der Walls equation describe a real gas behavior

Scientific Thinking - System under investigation : gas in a balloon - Description or Behavior of a system : Energy U, H, A, G Variables of a system : P, V, T - Correlation between behavior and variables of a system : T vs P, T vs. V, V vs. P - Modeling of a system : Mathematical expression for an ideal gas, Mathematical expression for a real gas - Comparison of a model with a system : Deviation of real gas from ideal gas - Revision of a model : More refined model for a real gas

Correlation between behavior and variables of a system Cause and Effect y = F(x) ; y is a function of x, x : controlled variable, y : observed variable

System - a part of the universe of interest to me - surrounded by the boundary to separate from the rest of the universe, called surroundings - Any interactions between the system and the surroundings? - What is exchanged between the system and the surroundings? - Any changes in the system due to the interactions and exchanges? - How do we describe these changes in a system with the interactions and exchanges?

System System / Boundary / Surroundings Open System Closed System Isolated System

Open System

Closed System

Isolated System

Description of a system composed of a pure gas Properties (variables or descriptors) of a pure gas system : - physical description: macroscopic properties of a gas system pressure P, volume V, and temperature T, chemical composition C, number of atoms or molecules, n - chemical description: µ (chemical potential) Intensive variables : T, P, µ(chemical potential) Extensive variables : V, n (# of moles of a gas) What about a molar volume (V/n)? If numerical values given to those descriptors, we know everything we need to know about the properties of the system. - the state of the system specified

Correlation between behavior and variables of a system Cause and Effect y = F(x) ; y is a function of x, x : controlled variable, y : observed variable

Properties of a gas : amount of gas n, temperature T, volume, V and pressure P Description of a gas at equilibrium Equilibrium of a system : - Thermal equilibrium : uniform T at each part of the system - Mechanical equilibrium : uniform P at each part of the system - Chemical equilibrium : uniform µ at each part of the system State of a gas: defined with numerical values given to amount of gas n, temperature T, volume V, and pressure P Equation of state of a gas : p = f(t, V, n), Equation of state of an ideal gas : pv = nrt

Equilibrium of a system : The word equilibrium means a state of balance. In an equilibrium state, there are no unbalanced potentials (or driving forces) within the system. Thermal equilibrium : T (thermal potential) is the same in every part of a system Mechanical equilibrium : P (mechanical potential) is the same in every part of a system Chemical equilibrium : µ (chemical potential) is the same in every part of a system P = f(t, V, n) pv = nrt for an ideal gas Gas A in a container P. V

Boyle s law pv = nrt

The Zeroth Law of Thermodynamics Consider two systems, A and B, in which the temperature of A is greater than the temperature of B. T A > T B - Each is a closed system. - No material transfer, but heat & work transfer across the boundary What happens to the temperature when A and B are brought together? - Heat flux from A to B due to temperature difference. Why? - Thermal energy transfer, or heat transfer from A to B till T A = T B - Two systems at thermal equilibrium

The Zeroth Law of Thermodynamics If two systems of any size are in thermal equilibrium with each other and a third system is in thermal equilibrium with one of them, then it is in thermal equilibrium with the other, too. If T A = T B and T B = T C, then T A = T C

State of a system The state of a system is dictated by what the state variables are, not by what they were, or how they got there. 1. Process 1 Process 2 Initial State Intermediate State Final State 2. Process 3 Initial State Final State

State of a system State Variable (Variable necessary to describe a state of system) : State function - precisely measurable physical property that characterizes the state of a system, independently of how the system was brought to that state. (path-independent) - must be inherently single-valued to characterize a state. State variables : pressure P, volume V, and temperature T, internal energy U, enthalpy H, Helmholtz free energy F and Gibbs free energy G and entropy S In the ideal gas law, the state of n moles of gas is precisely determined by these three state variables of P, V and T. PV = nrt

Equation of State of a gas PV = nrt = PV State Variables : P, T, V, n For any fixed amount of a pure gas, consider two state variables P and V. - For a given T, P and V can not be controlled independently from each other. - For any fixed amount of a pure gas, only two of the three state variables P, V, and T are truly independent. - Mathematical equation with which we can calculate the third state variable from the two known state variables : equation of state

Equation of State of a gas PV = nrt = PV The physical properties of a perfect gas are completely described by the amount of substance of which it is comprised, its temperature, its pressure and the volume which it occupies. These four parameters are not independent, and the relations between them are expressed in the ideal gas laws. The three historical gas laws Boyle s law, Charles law and Avogadro s principle are specific cases of the perfect gas equation of state, which is usually quoted in the form pv=nrt, where R is the gas constant.

Equation of State of a gas PV = nrt = PV The kinetic theory of gases is an attempt to describe the macroscopic properties of a gas in terms of molecular behavior (microscopic properties). Pressure is regarded as the result of molecular impacts with the walls of the container, and temperature is related to the average translational energy of the molecules. The molecules are considered to be of negligible size, with no attractive forces between them, travelling in straight lines. Molecules undergo perfectly elastic collisions, with the kinetic energy of the molecules being conserved in all collisions, but being transferred between molecules

Ideal Gas Law An ideal gas : - all collisions between gas molecules are perfectly elastic - there are no intermolecular attractive forces - no potential energy (no force field) - a collection of perfectly hard spheres which collide but which otherwise do not interact with each other - all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas : characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T).

Ideal Gas Law with Constraints : PV = nrt All the possible states of an ideal gas can be represented by a PVT surface as illustrated in the left. The behavior when any one of the three state variables is held constant is also shown.

Equation of State pv = nrt pv = nrt V/T = nr/p V = nrt/p pv = nrt Boyle s law Charles s law Avogadro s law at fixed n, T V/T = nr/p V/n = RT/p at fixed n, p at fixed p, T V = nrt/p R : Proportionality constant

Boyle s law pv = nrt

Charles s law V/T = nr/p

Avogadro s law V = nrt/p

Ideal Gas Law The relationship between T, P and V : deduced from kinetic theory of gases and is called the n = number of moles R = universal gas constant = 8.3145 J/mol K N = number of molecules k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 ev/k k = R/N A N A = Avogadro's number = 6.0221 x 10 23 /mol one mole of an ideal gas at STP occupies 22.4 liters. STP is used widely as a standard reference point for expression of the properties and processes of ideal gases. - standard temperature : freezing point of water, 0 C = 273.15 K - standard pressure : 1 atmosphere = 760 mmhg = 101.3 kpa - standard volume of 1 mole of an ideal gas at STP : 22.4 liters

SI base units Time is one of the so-called base units within the SI system, and so is length. Whereas volume can be expressed in terms of a length (for example, a cube has a volume l 3 and side of area l 2 ), we cannot define length in terms of something simpler. Similarly, whereas a velocity is a length per unit time, we cannot express time in terms of something simpler. In fact, just as compounds are made up of elements, so all scientific units are made up from seven base units: length, time, mass, temperature, current, amount of material and luminous intensity.

SI base units

Ideal Gas Law pv = nrt SI units: V (m 3 ), P (Pa), T (K) R = 8.314 J K -1 mol -1 1 J = 1 Nm, 1 N = 1 kg m s -2 1 Pa = 1 N/m 2 = 1 kg m s -2 /m 2 = 1 kg m -1 s -2 alternative units: V (L), (1 L = 1 dm 3 = 10-3 m 3 ), P (atm) R = 8.206 x 10-2 L atm K -1 mol -1 1 Pa = 1 N/m 2 = 10 5 bar = 9.8692 10 6 atm = 7.5006 10 3 torr = 145.04 10 6 psi

Partial Derivatives : How is one state variable affected when another state variable changes?

Derivatives : smoothly varying function y = f (x) We can draw a unique tangent line (a straight line whose slope matches the curve s slope) at each point on the curve. Recall that the slope of a line is defined as the amount y changes if x is changed by one; for example, the line y = 3x +6 has a slope of three. The mathematical definition of the derivative is

Partial Derivatives : F = function (x, y, z,..)? Change in F is expressed by df caused by a change in x by dx, a change in y by dy, a change in z by dz and so on. Total derivatives of a function of multiple variables F = function (x, y, z,..)? the derivative of the function F taken w.r.t. one variable at a time with the other variables held constant the derivative of the function F taken w.r.t. x only with y, z and so on treated as constants : partial derivatives

Partial Derivative When there is more than one variable in a function, it is often useful to examine the variation of the function with respect to one of the variables with all the other variables constrained to stay constant. This is the purpose of a partial derivative. the partial derivative with respect to x, holding y as a constant : the partial derivative with respect to y holding x as a constant : :

Partial Derivatives and Ideal Gas Law How is one state variable affected when another state variable changes? V = f(t,p) V T P

Partial Derivatives and Ideal Gas Law How does the P varies w.r.t. T, assuming constant n and V? PV = nrt, P = nrt/v

Non-ideal Gas - Ideal gas : PV= nrt - Real gas : molecular interaction, molecular size Non-ideality of real gas Deviation of real gas from ideal gas Quantifying non-ideality

Ideal Gas Law An ideal gas : - all collisions between gasmolecules are perfectly elastic - there are no intermolecular attractive forces - no potential energy (no force field) - a collection of perfectly hard spheres which collide but which otherwise do not interact with each other - all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas : characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T).

Ideal Gas Law Gas Kinetic Theory

Non-ideal Gas - Ideal gas : PV= nrt - Real gas : molecular interaction, molecular size Why Non-ideality for real gas? Deviation of real gas from ideal gas Quantifying non-ideality

Scientific Thinking - System under investigation : gas in a balloon - Description or Behavior of a system : Energy, # of moles of gas, pressure, volume, temperature - Variables of a system : pressure, volume, temperature - Correlation between behavior and variables of a system : T vs P, T vs. V, V vs. P - Modeling of a system : Mathematical expression for an ideal gas, Mathematical expression for a real gas - Comparison of a model with a system : Deviation of real gas from ideal gas - Revision of a model : More refined model for a real gas

Description of Non- ideal Gas - Start with the equation of state of an ideal gas : PV= nrt - Different behavior of a non-ideal gas from an ideal gas - To develop the equation of state of a non-ideal gas, modify the equation of state of an Ideal gas, considering molecular interaction, molecular size effect of a non-ideal gas van der Waals gas equation - Deviation of real gas from ideal gas - Analysis of non-ideality using van der Waals gas equation

Non- ideal Gas - Intermolecular interaction Coulomb's Law Like charges repel, unlike charges attract. The electric force acting on a point charge q 1 as a result of the presence of a second point charge q 2 is given by Coulomb's Law: where ε 0 = permittivity of space - Finite particle size

Non- ideal Gas Compressibility Factor

Non- ideal Gas Modification of the ideal gas equation : the van der Waals equation How can the ideal gas law be modified to yield an equation that will represent the experimental data more accurately? In the ideal gas law, namely the prediction that under any finite pressure the volume of the gas is zero at the absolute zero of temperature : On cooling, real gases liquefy and ultimately solidify ; after liquefaction the volume does not change very much. We can arrange the new equation so that it predicts a finite, positive volume for the gas at 0 K by adding a positive constant b to the ideal volume

Since Eq. (3.3) requires Z to be a linear function of pressure with a positive slope b/rt, it cannot possibly fit the curve for nitrogen in Fig. 3. 1, which starts from the origin with a negative slope. However, Eq. (3.3) can represent the behavior of hydrogen. In Fig. 3. 1 the dashed line is a plot of Eq. (3.3) fitted at the origin to the curve for hydrogen. In the low-pressure region the dashed line represents the data very well.

We have already noted that the worst offenders in the matter of having values of Z less than unity are methane and carbon dioxide, which are easily liquefied. Thus we begin to suspect a connection between ease of liquefaction and the compressibility factor, and to ask why a gas liquefies.

Consider two small volume elements V 1 and V 2 in a container of gas (Fig. 3.3). Suppose that each volume element contains one molecule and that the attractive force between the two volume elements is some small value f. If another molecule is added to V 2, keeping one molecule in V 1, the force acting between the two elements should be 2f ; addition of a third molecule to V 2 should increase the force to 3f, and so on. The force of attraction between the two volume elements is therefore proportional to C 2, the concentration of molecules in V 2. If at any point in the argument, the number of molecules in V 2 is kept constant and molecules are added to V 1, then the force should double and triple, etc. The force is therefore proportional to C 1, the concentration of molecules in V 1. Thus, the force acting between the two elements can be written as follows. Force is proportional to the product of [C 1 x C 2 ]. Since the concentration in a gas is everywhere the same, C 1 = C 2 = C, and so, force is proportional to C 2. But C = n/v = 1/V m ; consequently, force is proportional to 1/V m2.

Because of the attractive forces between the molecules, the pressure is less than that given by Eq. (3.4) by an amount proportional to 1/V m2, so a term is subtracted from the right-hand side to yield where a is a positive constant roughly proportional to the energy of vaporization of the liquid.

Ideal gas law : - molecules of a gas as point particles with perfectly elastic collision - valid for dilute gases, but gas molecules are not point masses van der Waals equation of state ; - A modification of the ideal gas law proposed by Johannes D. van der Waals in 1873 to take into account molecular size and molecular interaction forces. Constants a and b : - positive values and characteristic of the individual gas - gas-specific properties, different values for different gases - The van der Waals equation of state approaches the ideal gas law PV=nRT as the values of these constants approach zero.

van der Waals Equation of State Constant a : correction for the intermolecular forces Constant b : correction for finite molecular size - its value is the volume of one mole of the atoms or molecules - could be used to estimate the radius of an atom or molecule, modeled as a sphere.

Ideal Gas Law An ideal gas : - all collisions between atoms or molecules are perfectly elastic - there are no intermolecular attractive forces - a collection of perfectly hard spheres which collide but which otherwise do not interact with each other - no potential energy (no force field) - all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas : characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T).

Ionic Bonding Regular stacking of Na + and Cl ions in solid NaCl, which is indicative of the nondirectional nature of ionic bonding. 6 Na + surround each Cl -, and 6 Cl - surround each Na +. The ionic bond is the result of the coulombic attraction.

Ionic Bonding Net bonding force curve for a Na + Cl pair showing an equilibrium bond length of a 0 = 0.28 nm. - - F = E/ r Z 1, Z 2 = Number of electrons removed or added during ion formation e = Electron Charge, a = Interionic seperation distance ε = Permeability of free space (8.85 x 10-12 C 2 /Nm 2 ) -

Ionic Bonding Intermolecular interaction Energy minimum energy most stable Energy balance of attractive and repulsive terms A E N = E A + E R = + r B r n Repulsive energy E R Interatomic separation r Net energy E N Attractive energy E A

Intermolecular Forces and Potentials Intermolecular interaction F = E/ r

Non- ideal Gas

Non- ideal Gas Intermolecular force? : condensation, liquefaction at high P Ideal Gas Liquefaction of a real gas No liquefaction of an ideal gas Requirements for liquefaction of a real gas?

Non- ideal Gas D D isotherm D D D E D Phase : a physically distinctive form of matter, such as a solid, liquid, gas or plasma. At point C A phase of matter is characterized by having relatively uniform chemical and physical properties. Phases are different from states of matter. The states of matter (e.g., liquid, solid, gas) are phases, but matter can exist in different phases yet the same state of matter. For example, mixtures can exist in multiple phases, such as an oil phase and an aqueous phase.

Phase Equilibria in water

Pc At Critical Point, Molar volume of liquid = Molar volume of gas Critical Point Critical Temperature Tc Critical Pressure Pc Critical Volume Vc Molar volume of liquid Molar volume of gas Vc

Critical point The liquid-vapour equilibrium curve has a top limit labelled as C in the phase diagram. This is known as the critical point. The temperature and pressure corresponding to this are known as the critical temperature and critical pressure. If you increase the pressure on a gas (vapor) at a temperature lower than the critical temperature, you will eventually cross the liquid-vapour equilibrium line and the vapor will condense to give a liquid.

Critical point What if your temperature was above the critical temperature? There wouldn't be any line to cross! Above the critical temperature, it is impossible to condense a gas into a liquid just by increasing P. Instead of existing as separate liquid and vapor phases, the system exists as a single supercritical fluid phase. The transition to this supercritical fluid phase is continuous and is therefore not a true phase change. All you get is a highly compressed gas. At the critical state, the particles have too much energy for the intermolecular attractions to hold them together as a liquid. The critical temperature obviously varies from substance to substance and depends on the strength of the attractions between the particles. The stronger the intermolecular attractions, the higher the critical temperature.

Cryophorus : an instrument that illustrates the freezing of water by its own evaporation A cryophorus is a glass container containing liquid water and water vapor. It is used to demonstrate rapid freezing by evaporation. A typical cryophorus has a bulb at one end connected to a tube of the same material. When the liquid water is manipulated into the bulbed end and the other end is connected to a vacuum pump, the gas pressure drops as it is cooled. The liquid water begins to evaporate, producing more water vapor. Evaporation causes the water to cool rapidly to its freezing point and it solidifies suddenly. Finally, a piece of ice disappears as a result of ice sublimation to water vapor. Vacuum pump Water droplet in a bottle under vacuum

Cryophorus : A cryophorus is a glass container containing liquid water and water vapor. It is used to demonstrate rapid freezing by evaporation. When the liquid water is manipulated into the bulbed end and the other end is connected to a vacuum pump, the gas pressure drops as it is cooled. The liquid water begins to evaporate, producing more water vapor. Evaporation causes the water to cool rapidly to its freezing point and it solidifies suddenly. Finally, a piece of ice disappears as a result of ice sublimation to water vapor. Vacuum pump Water droplet in a bottle under vacuum

Non- ideal Gas Intermolecular force vs. critical temperature

Intermolecular Forces and Potentials

Intermolecular Forces and Potentials Force Potential Energy V(z)

van der Waals Equation of State : particle size b

Other Equations of State of Non-Ideal Gas

van der Waals Equation of State : particle size b

Non- ideal Gas : van der Waals Equation of State

van der Waals Equation : intermolecular interaction a

van der Waals Equation of State : intermolecular interaction Real P are less than ideal value by the attraction force from interior of the gas (internal pressure). Pressure on wall depends on both collision frequency and collision force, both related to the molar concentration (n/v = 1/V m ), so P reduced by a correction term which is square of molar conc. a/v m2.

Non- ideal Gas : van der Waals Equation of State

Residual Volume Ideal gas : PV = nrt, PV m, ideal = RT, V m, ideal = RT/P (R T)/(P V m ) = V m, ideal /V m Residual volume = V m V m, ideal

Residual Volume

Residual Volume = Vm Vm, ideal

Residual Volume

Non- ideal Gas : van der Waals Equation of State

Non-ideal Gas : how to express non-ideality? N moles of Gas in a balloon P, T, V Ideal gas : PV = nrt, PV m = RT Compressibility factor Z (T, P) = PV m /RT = P/(RT/V m ) = P/P ideal

Compressibility factor Z (T, P) = PV m /RT = P/(RT/V m ) = P/P ideal Compression Factor = Molar volume of gas/molar volume of perfect gas May it be different from1?

Non- ideal Gas : how to express non-ideality? Compressibility factor Z, Ideal gas : PV= nrt

Non- ideal Gas : how to express non-ideality? Compressibility factor Z, Ideal gas : PV= nrt Homework : Compressibility factor Z for van der Walls gas? Due 19 March

Virial Equation of State

Compressibility factor Z = P V m /RT at moderate pressure (low P and large V), b/v m < 1 since for x << 1.

This equation expresses Z as a function of temperature and molar volume. It would be preferable to have Z as a function of temperature and pressure ; however, this would entail solving the van der Waals equation for V as a function of T and p, then multiplying the result by p/rt to obtain Z as a function of T and p. At P = 0, all of the higher terms drop out and this derivative reduces simply to

Compressibility factor Z, Z as a function of T and V Z as a function of T and P

Non- ideal Gas

where the derivative is the initial slope of the Z versus p curve. If b > a/rt, the slope is positive ; the size effect dominates the behavior of the gas. On the other hand, if b < a/r T, then the initial slope is negative ; the effect of the attractive forces dominates the behavior of the gas. Thus the van der Waals equation, which includes both the effects of size and of the intermolecular forces, can interpret either positive or negative slopes of the Z versus p curve. In interpreting Fig. 3.2, we can say that at 0 C the effect of the attractive forces dominates the behavior of methane and carbon dioxide, while the molecular size effect dominates the behavior of hydrogen.

Non- ideal Gas Boyle Temperature at which (dz/dp) at P=0 approaches zero. At T Boyle, b=a/rt, then T Boyle =a/rb

Non- ideal Gas If the temperature is low enough, the term a/rt will be larger than b and so the initial slope of Z versus p will be negative. As the temperature increases, a/rt becomes smaller and smaller ; if the temperature is high enough, a/rt becomes less than b, and the initial slope of the Z versus p curve becomes positive. Finally, if the temperature is extremely high, Eq. (3.9) shows that the slope of Z versus p must approach zero. At some intermediate temperature T B, the Boyle temperature, the initial slope must be zero. The condition for this is given by Eq. (3.9) as b a/rt B = 0. T B =a/rb

Non- ideal Gas At the Boyle temperature the Z versus p curve is tangent to the curve for the ideal gas at p = 0 and rises above the ideal gas curve only very slowly. In Eq. (3.8) the second term drops out at T B, and the remaining terms are small until the pressure becomes very high. Thus at the Boyle temperature the real gas behaves ideally over a wide range of pressures, because the effects of size and of intermolecular forces roughly compensate. Z = 1, (dz/dp) p=0 = 0

Non- ideal Gas The data in Table 3.2 make the curves in Fig. 3.2 comprehensible. All of them are drawn at 0 o C. Thus hydrogen is above its Boyle temperature and so always has Z-values greater than unity. The other gases are below their Boyle temperatures and so have Z-values less than unity in the low-pressure range.

van der Waals Equation of State intermolecular interaction vs. melting point

Virial Equation of State

Other Equations of State

Non- ideal Gas The decrease in volume over a wide range in which the pressure remains at the constant value P e. At a somewhat higher temperature the behavior is qualitatively the same, but the range of volume over which condensation occurs is smaller and the vapor pressure is larger.

Non- ideal Gas highly compressed gaseous state of the substance Liquid Liquid & Vapor in equilibrium Gas Two - phase region and continuity of states A phase is a region of uniformity in a system. This means a region of uniform chemical composition and uniform physical properties. Thus a system containing liquid and vapor has two regions of uniformity. In the vapor phase, the density is uniform throughout. In the liquid phase, the density is uniform throughout but has a value different from that in the vapor phase.

Non- ideal Gas When V is very large this equation approximates the ideal gas law, since V is very large compared with b and a/v 2 is very small compared with the first term. This is true at all temperatures. At high temperatures, the term a/v 2 can be ignored, since it is small compared with RT/(V-b). Isotherms of the van der Waals gas At lower temperatures and smaller volumes, none of the terms in the equation may be neglected. The result is rather curious. At the temperature T c the isotherm develops a point of inflection, point E. At still lower temperatures, the isotherms exhibit a maximum and a minimum.

Non- ideal Gas Cubic equation, it may have three real roots for certain values of pressure and temperature. There is a certain maximum pressure P c and a certain maximum temperature T c, at which liquid and vapor can coexist. Isotherms of the van der Waals gas This condition of temperature and pressure is the critical point and the corresponding volume is the critical volume V c. At the critical point the three- roots are all equal to V c ;. The cubic equation can be written in terms of its roots V, V, V. At the critical point V' = V" = V'" = V c, so that the equation becomes (V - V c ) 3 = O.

Non- ideal Gas At the critical point Isotherms of the van der Waals gas

van der Waals Equation of State P V Below critical temperature

Law of Corresponding States Reduced variables P r T r V r

Law of Corresponding States

Law of Corresponding States P P V P P vs V plot at critical temperature

Law of Corresponding States

Law of Corresponding States Reduced variables P r T r V r The properties of all gases are the same if we compare them under the same conditions relative to their critical point.

Law of Corresponding States PV = nrt No gas-specific properties, a, b : gas-specific properties, different values for different gases

Law of Corresponding States a, b : gas-specific properties, different values for different gases

Law of Corresponding States

Law of Corresponding States Gases in states with the same values of T r and P r deviate from the ideality to the same extent; i.e. the values of Z = PV/(RT) are approximately same for different gases at the same value of T r and P r. In other words, the reduced volume of all gases are the same when the gases are at the same reduced pressures and temperatures.

Law of Corresponding States Two gases at the same reduced temperature and under the same reduced pressure are in corresponding states. By the law of corresponding states, they should both occupy the same reduced volume. For example, argon at 302 K and under 16 atm pressure, and ethane at 381 K and under 18 atm are in corresponding states, since each has T r = 2 and P r = 1/3.

Law of Corresponding States At very low pressures (P r <<1), the gases behave as an ideal gas regardless of temperature. At high temperature (T r > 2), ideal gas behaviour can be assumed with good accuracy regardless of pressure except when (P r >> 1). The deviation of a gas from ideal gas behaviour is greatest in the vicinity of the critical point.

Law of Corresponding States It may be seen from the chart that the value of the compressibility factor at the critical state is about 0.25. Note that the value of Z obtained from Van der waals equation of state at the critical point, which is higher than the actual value.

Law of Corresponding States R = 0.082 L atm mol -1 K -1

Law of Corresponding States