Calorimetry: Problem Solving with Heat Exchanges (method 1)

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1 Calorimetry: Problem Solving with Heat Exchanges (method 1) Main Concept: Conservation of Energy Q = 0 (sum of all heat flows into and out of system =0) Sign Convention: heat enters a system is + heat leaves a system is T = T f T i

2 Calorimetry: Problem Solving with Heat Exchanges (method 2) Main Concept: Conservation of Energy OR Qgain Qloss Keep all heats as positive quantities

3 Calorimetry: Problem Solving with Heat Exchanges Steps: 1. Identify all phase change pts 2. Apply (either Q=mcT or Q=mL) for each processes separately. (don t apply Q=mcT across ph. changes!) 3. Use Q ALL = 0 and follow sign convention or just do Qgain Qloss

4 Calorimetry (example 17.8) Ice initially at -20 o C note 0.25kg Cola Initially at 25 o C Question: How much ice needed so that the final mixture is all liquid water with a temperature of 0 o C?

5 Mechanisms of Heat Transfer #1: Conduction H dq dt T ka H T L C (+H is in the dir. of decreasing T) H heat current [J/s] (heat flow rate) k thermal conductivity / (characteristic of the material) R = L/k thermal resistance (larger is better) W m K

6 Thermal Resistivity (additive R values) TH H 1 2 TM H ( heat current) ka( TH TL)/ L A( TH TL)/ R where R L / k TL For a composite system, we have the following: A TH T A T M M TL H1 H 2 R R 1 and 2 Now, by conservation of energy, we need to have H1 H2 H Re-arranging and adding the two equations gives: T T H M T M T L HR1 A H R1 R2 A TH T TH TL H A R1 R2 HR2 A So, for composite system, R is additive. L

7 Mechanisms of Heat Transfer Protective tile for the space shuttle has both low values of k and c!

8 Mechanisms of Heat Transfer #2: Convection Heating by moving large amounts of hot fluid, usually water or air. Heating element in the tip warms surrounding water. Heat is transferred by convection of the warm water movement.

9 Mechanisms of Heat Transfer #3: Radiation Infrared lamps, hot objects, a fireplace, standing near a running furnace these are all objects heating others by broadcasting EM radiation. H Ae T 4 (Stefan-Boltzmann Law) surface area of object at T e emissivity [0,1] (effectiveness of surface in emitting EM radiation ) Stefan-Boltzmann constant (a fundamental physical constant) Camera sensitive to these radiation can be used to take this picture.

10 Mechanisms of Heat Transfer Radiation and Absorption The environment around an object at a given T also radiates electromagnetic energy and the radiating object will absorb some of this energy. In general, the absorption will again depends on the surface properties of the object, i.e., the same A, e, and. Now, if the surrounding environment is at T s, the net heat current radiated by the object will be, H AeT AeT Ae( T T ) net s s (radiate) (absorb)

11 PHYS 262/266 Geroge Mason University Prof. Paul So

12 Chapter 18: Thermal Properties of Matter Topics for Discussion Equations of State Ideal Gas Equation PV Diagrams Kinetic-Molecular Model of an Ideal Gas Heat Capacities Distribution of Molecular Speeds Phases of Matter

13 Equations of State State Variables physical variables describing the macroscopic state of the system: P, V, T, n (or m) Equation of State a mathematical relationship linking these variables

14 The Ideal Gas Equation Properties of a gas is studied by varying the macroscopic variables: P, V, T, n and observing the result. Observations: 1. P 1 V e.g. air pump 2. V T e.g. hot air balloon 3. P T e.g. hot closed spray can 4. V n e.g. birthday balloon

15 Ideal Gas Law (summary) By putting all these observations together, we have PV nrt R Universal Gas Constant (R = J/mol K) (This is an important example of an Equation of State for a gas at thermal equilibrium.) An Ideal Gas (diluted): No molecular interactions besides elastic collisions Molecular volume <<< volume of container Most everyday gases ~ Ideal!

16 The Ideal Gas Law Important Notes: The relationship V vs. T (at cont P) & P vs. T (at cont V) are linear for all diluted gases. diluted gas ~ Ideal They both extrapolate to a single zero point (absolute zero). T has to be in K! P vs. const V V vs. const P T = o C Pressure Volume Temperature ( o C) P V nr V nr P T T

17 The Ideal Gas Law (alternative form) Instead of the number of moles (n), one can specify the amount of gas by the actual number of molecules (N). N = n N ( N molecules/ mole) A A where N A is the # of molecules in a mole of materials (Avogadro s number). N PV nrt PV RT NkT N where k is the Boltzmann constant, k R N / A A J molecule K

18 Example 18.1 (V at STP) What is the volume of a gas (one mole) at Standard Temperature and Pressure (STP)? STP: T = 0 o C = K 5 P = 1 atm = Pa nrt PV nrt V P (1 mole)(8.314 J / molk)( K) m Pa L

19 Typical Usage for the Ideal Gas Law For a fixed amount of gas (nr=const) PV nr const T So, if we have a gas at two different states 1(before) and 2 (after), their state variables are related simply by: PV T PV T We can use this relation to solve for any unknown variables with the others being given.

20 Example 18.2 In an automobile engine, a mixture of air/gasoline is being compressed before ignition. Typical compression ration 1 to 9 Initial P = 1 atm and T = 27 o C note Find the temperature of the compressed gas if we are given the pressure after compression to be 21.7atm.

21 The Ideal Gas Law (graphical view) P,V,T relationship in the Ideal Gas Law can be visualize graphically as a surface in 3D. P nrt V

22 PV Diagrams 2D projections of the previous 3D surface. Evolution of a gas at constant T will move along these curves called isotherms. Gives P vs. V at a various T: P ( nrt ) 1 V