Unit 05 Kinetic Theory of Gases Unit Concepts: A) A bit more about temperature B) Ideal Gas Law C) Molar specific heats D) Using them all Unit 05 Kinetic Theory, Slide 1
Temperature and Velocity Recall: T <KE TRAN > ½ kt =<½mv x2 > Boltzman s Constant = 1.38X10-23 J/K v rms =[3kT/m] 0.5 Unit 05 Kinetic Theory, Slide 2
Example 5.1: Gas Velocity What is the average velocity of a nitrogen (N 2 ) molecule in earth s atmosphere? (T=287K) What is the average velocity of a hydrogen (H 2 ) molecule in earth s atmosphere? (T=287K) Unit 05 Kinetic Theory, Slide 3
Variation with Temperature v rms = v peak 4 m f ( v) ( ) 2kT 3 2 2 2 mv 2kT v e Unit 05 Kinetic Theory, Slide 4
Variation with Mass Unit 05 Kinetic Theory, Slide 5
Universal Gas Constant k*n av = = (1.38X10-23 J/K)*(6.02X10 23 particle/mole) = 8.31 J/K mol = R (universal gas constant) K*N = n*r Unit 05 Kinetic Theory, Slide 6
Example 5.2: Ideal Gas Law A 0.5m 3 cube holds an ideal gas at 20 o C and pressure = 2bar. How many moles of gas are in the cube? Unit 05 Kinetic Theory, Slide 7
Example 5.3: Ideal Gas Law You have 0.2m 3 cube of an ideal gas at 24 o C and 1 atmosphere. You heat the gas to 100oC with a constant volume. What is the final pressure? Unit 05 Kinetic Theory, Slide 8
Example 5.4: Ideal Gas Law 2 moles of an ideal gas are initially at 300K in a 2m 3 container. How much work is done decreasing the volume to 1m 3 at constant T? Unit 05 Kinetic Theory, Slide 9
First Law of Thermodynamics DE int = Q - W Q DE int + W Heat in must Increase Temp Do Work Unit 05 Kinetic Theory, Slide 10
Example 5.5: First Law 6J of heat is added to an ideal gas contained in a cylinder with a movable piston. The piston rises doing 2J of work. What is the change in internal energy of the gas? Let s say instead that an electric motor pushes the piston down doing 2J of work. What is the change in internal energy of the gas? Unit 05 Kinetic Theory, Slide 11
Molar Specific Heats Combine: - 1 st Law and - Ideal Gas Law to predict the molar specific heat capacities for gases Unit 05 Kinetic Theory, Slide 12
Molar Heat Capacity Constant Volume Unit 05 Kinetic Theory, Slide 13
Molar Heat Capacity Constant Volume Monatomic three degrees of freedom (3 translation) c v = 3/2R Diatomic five degrees of freedom (3 translation + 2 rotation) c v = 5/2R Complex seven degrees of freedom (3 translation + 2 rotation + 2 vibration) c v = 7/2R Unit 05 Kinetic Theory, Slide 14
Polyatomic C v Unit 05 Kinetic Theory, Slide 15
Specific Heat - Solids No rotation. Only vibration in all three directions. E = 1 2 mv x 2 + 1 2 mv y 2 + 1 2 mv z 2 + 1 2 kx2 + 1 2 ky2 + 1 2 kz2 c v =6/2 R=3R Unit 05 Kinetic Theory, Slide 16
Specific Heat - Solids Unit 05 Kinetic Theory, Slide 17
C v and C p Unit 05 Kinetic Theory, Slide 18
Question In which of the following situations will the internal energy, E int, of a system change the most: A. 100 J of heat is withdrawn from the system while the volume of the system is doubled B. 100J of heat is withdrawn from the system while the volume of the system is remains fixed. C. 100J of heat is withdrawn from the system while the volume of the system is cut in half. Unit 05 Kinetic Theory, Slide 19
Summary c v = f/2r (constant volume) E int = nc v T (for all cases) c p = (f/2 + 1)R = c v + R (constant pressure) Unit 05 Kinetic Theory, Slide 20
Example 5.6: Heat Capacity The heat capacity at constant volume of a certain amount of a diatomic gas is 60.5 J/K. How many moles of the gas are present? What is the internal energy of the gas at T = 300 K? What is the heat capacity at constant pressure of the gas? Unit 05 Kinetic Theory, Slide 21
Example 5.7: Adiabatic Expansion 2liters of air at 2atm and 20 o C expands adiabatically to 4liters. What is the final temperature? Unit 05 Kinetic Theory, Slide 22
Example 5.8: Gas Cycle An ideal monatomic gas is compressed isothermally to 1/5 it s original volume. It is then expanded isobarically back to it s original volume. Finally it is cooled isochorically to its original pressure. In terms of p 1 and V 1, how much work is done? Unit 05 Kinetic Theory, Slide 23
Example 5.9: Gas Cycle II 2 moles monatomic gas is initially at standard pressure and temperature (STP) and undergoes the following cycle: 1->2: isobaric expansion to 4 times the original volume 2->3: adiabatic cooling back to the original temperature 3->1: isothermal process back to the original state In terms of p 1 and V 1, determine DE int, Q and W for each step. Unit 05 Kinetic Theory, Slide 24