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Constant-Volume Process Constant-Volume Process . A constant-volume process is called an isochoric process. Consider the gas in a closed, rigid container. Warming the gas with a flame will raise its pressure without changing its volume. © 2017 Pearson Education, Inc. Slide 18-1 Example 1 A constant-volume gas thermometer is placed in contact with a reference cell containing water at the triple point (T = 0.01ºC). After reaching equilibrium, the gas pressure is recorded as 55.78 kPa. The thermometer is then placed in contact with a sample of unknown temperature. After the thermometer reaches a new equilibrium, the gas pressure is 65.12 kPa. What is the temperature of this sample? © 2017 Pearson Education, Inc. Slide 18-2 QuickCheck A cylinder of gas has a frictionless but tightly sealed piston of mass M. A small flame heats the cylinder, causing the piston to slowly move upward. For the gas inside the cylinder, what kind of process is this? A. Isochoric B. Isobaric C. Isothermal D. Adiabatic © 2017 Pearson Education, Inc. Slide 18-3 Constant-Pressure Process . A constant-pressure process is called an isobaric process. Consider a cylinder of gas with a tight-fitting piston of mass M that can slide up and down but seals the container. In equilibrium, the gas pressure inside the cylinder is © 2017 Pearson Education, Inc. Slide 18-4 QuickCheck A cylinder of gas has a frictionless but tightly sealed piston of mass M. The gas temperature is increased from an initial 27ºC to a final 127ºC. What is the final-to-initial volume ratio Vf /Vi? A. 1.50 B. 1.33 C. 1.25 D. 1.00 © 2017 Pearson Education, Inc. Slide 18-5 Example 2 The two cylinders below contain ideal gases and are both sealed by a frictionless piston of mass M. How does the pressure of gas 2 compare to that of gas 1? Suppose gas 2 is heated to 80ºC. What happens to the pressure and volume? © 2017 Pearson Education, Inc. Slide 18-6 QuickCheck A cylinder of gas floats in a large tank of water. It has a frictionless but tightly sealed piston of mass M. Small masses are slowly placed onto the top of the piston, causing it to slowly move downward. For the gas inside the cylinder, what kind of process is this? A. Isochoric B. Isobaric C. Isothermal D. Adiabatic © 2017 Pearson Education, Inc. Slide 18-7 Constant-Temperature Process . A constant-temperature process is called an isothermal process. Consider a piston being pushed down to compress a gas. Heat is transferred through the walls of the cylinder to keep T fixed, so that . The graph of p versus V for an isotherm is a hyperbola. © 2017 Pearson Education, Inc. Slide 18-8 QuickCheck In an isothermal expansion of an ideal gas, the amount of heat that flows into the gas A. is greater than the amount of work done by the gas. B. equals the amount of work done by the gas. C. is less than the amount of work done by the gas, but greater than zero. D. is zero. © 2016 Pearson Education, Inc. QuickCheck A gas follows the process shown. What is the final-to-initial temperature ratio Tf /Ti? A. 2 B. 4 C. 8 D. 16 © 2017 Pearson Education, Inc. Slide 18-10 Example 3 A gas at 2.0 atm pressure and temperature of 200ºC is first expanded isothermally until its volume has doubled. It then undergoes an isobaric compression until it returns to its original volume. First show this process on a pV-diagram. Then find the final temperature and pressure. © 2017 Pearson Education, Inc. Slide 18-11 Internal energy of an ideal gas • The internal energy of an ideal gas depends only on its temperature, not on its pressure or volume. • The temperature of an ideal gas does not change during a free expansion. © 2016 Pearson Education Inc. Heat capacities of an ideal gas • CV is the molar heat capacity at constant volume. • To measure CV, we raise the temperature of an ideal gas in a rigid container with constant volume, ignoring its thermal expansion. © 2016 Pearson Education Inc. Heat capacities of an ideal gas • Cp is the molar heat capacity at constant pressure. • To measure Cp, we let the gas expand just enough to keep the pressure constant as the temperature rises. © 2016 Pearson Education Inc. Relating Cp and CV for an ideal gas • To produce the same temperature change, more heat is required at constant pressure than at constant volume since is the same in both cases. • This means that Cp > CV. • Cp = CV + R. • R is the gas constant R = 8.314 J/mol ∙ K. © 2016 Pearson Education Inc. CP and CV . Note that for all ideal gases where R = 8.31 J/mol K is the universal gas constant. © 2017 Pearson Education, Inc. Slide 19-16 QuickCheck An ideal gas begins in a thermodynamic state a. When the temperature of the gas is raised from T1 to a higher temperature T2 at a constant volume, a positive amount of heat Q12 flows into the gas. If the same gas begins in state a and has its temperature raised from T1 to T2 at a constant pressure, the amount of heat that flows into the gas is A. greater than Q12. B. equal to Q12. C. less than Q12, but greater than zero. D. zero. QuickCheck 1 mol of air has an initial temperature of 20ºC. 200 J of heat energy are transferred to the air in an isochoric process, then 200 J are removed in an isobaric process. Afterward, the air temperature is A. < 20ºC B. = 20ºC C. > 20ºC D. Not enough information is given to answer the question. © 2017 Pearson Education, Inc. Slide 19-18 The ratio of heat capacities • The ratio of heat capacities is: • For monatomic ideal gases, • For diatomic ideal gases, © 2016 Pearson Education Inc. Example 4 Example 4 – Three moles of 02 gas are at 20.0ºC. 600 J of heat energy are transferred to the gas at constant pressure, then 600 J are removed at constant volume. What is the final temperature? Show the process on a pV diagram. © 2017 Pearson Education, Inc. Slide 19-20 In-class Activity #1 A typical dorm room or bedroom contains about 2500 moles of air. Find the change in the internal energy of this air when it is cooled from 35.0ºC to 26.0ºC at a constant pressure of 1.00 atm. Treat the air as an ideal gas with γ = 1.400. © 2017 Pearson Education, Inc. Slide 19-21 Adiabatic processes for an ideal gas • In an adiabatic process, no heat is transferred in or out of the gas, so Q = 0. • Shown is a pV-diagram for an adiabatic expansion. • As the gas expands, it does positive work W on its environment, so its internal energy decreases, and its temperature drops. • Note that an adiabatic curve at any point is always steeper than an isotherm at that point. © 2016 Pearson Education Inc. QuickCheck An ideal gas is taken around the cycle shown in this p-V diagram, from a to c to b and back to a. Process c b is adiabatic. For process c b, A. Q > 0, W > 0, ∆U = 0 B. Q > 0, W > 0, ∆U > 0 C. Q = 0, W > 0, ∆U < 0 D. Q = 0, W < 0, ∆U > 0 E. Q < 0, W < 0, ∆U = 0 QuickCheck When an ideal gas is allowed to expand isothermally from volume V1 to a larger volume V2, the gas does an amount of work equal to W12. If the same ideal gas is allowed to expand adiabatically from volume V1 to a larger volume V2, the gas does an amount of work that is A. less than W12. B. greater than W12. C. equal to W12. D. either A or B, depending on the ratio of V2 to V1. E. any of A, B, or C, depending on the ratio of V2 to V1. Adiabatic Processes . An adiabatic process is one for which where . Adiabats are steeper than hyperbolic isotherms, so the temperature falls during an adiabatic expansion and rises during an adiabatic compression. © 2017 Pearson Education, Inc. Slide 19-25 QuickCheck A gas in a container expands rapidly, pushing the piston out. The temperature of the gas A. Rises. B. Is unchanged. C. Falls. D. Can’t say without knowing more. © 2017 Pearson Education, Inc. Slide 19-26 QuickCheck A gas in a container expands rapidly, pushing the piston out. The temperature of the gas falls. This is because A. The gas pressure falls. B. The gas density falls. C. Heat energy is removed. D. Work is done. © 2017 Pearson Education, Inc. Slide 19-27 Example 5 Air containing gasoline vapor is admitted into the cylinder of an internal combustion engine at 1.00 atm pressure and 30ºC. The piston rapidly compress the gas from 500 cm3 to 50 cm3 (compression ratio 10). (a) What are the final temperature and pressure of the gas? (b) Show the compression process on a pV diagram. (c) How much work is done to compress the gas? © 2017 Pearson Education, Inc. Slide 19-28 In-class Activity #2 The compression ratio of a diesel engine is 15.0 to 1. If the initial pressure is 1 atm and the initial temperature is 300 K, find the final pressure and the temperature after adiabatic compression. Also, how much work does the gas so during the compression if the initial volume of the cylinder is 1.00L? © 2017 Pearson Education, Inc.
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