Posted: September 20th, 2022

CW03-04 Exercise 4.1 Simple Changes in State

CW03-04
Exercise 4.1 Simple Changes in State
Consider one mole of an ideal gas with a molar heat capacity of Cv = 5R/2. This gas is
expanded from an inital state of T=300.0K and Volume of 10.0L to a final Volume of 20.0L
A. Consider a free expansion (pop = 0) of this gas between the endpoints defined above.
i What is the final pressure?
ii What is Work?
iii What is the change in Energy of the system, ∆U?
iv What is the change in Entropy of the system, ∆S?
v What is the change in Entropy of the Universe, ∆S Universe?
B. Consider the change in state of a single-stage isothermal expansion that creates the most usable
(most negative) Work between the endpoints defined above.
i What is the final pressure?
ii What is Work?
iii What is the change in Energy of the system, ∆U?
iv What is the change in Entropy of the system, ∆S?
v What is the change in Entropy of the Universe, ∆S Universe?
C. Consider the adiabatic (Q = 0) single-stage (constant external pressure) change in state that
creates the most usable (most negative) Work between the endpoints defined above.
i What is the final pressure?
ii What is Work?
iii What is the change in Energy of the system, ∆U?
iv What is the change in Entropy of the system, ∆S?
v What is the change in Entropy of the Universe, ∆S Universe?
Exercise 4.2 A Non-Ideal Gas
Consider a non-ideal gas that obeys the following equation of state:
pV = RT + aT2
where a is a constant specific to the gas.
Derive an expression for the Work performed by this gas in a reversible isothermal expansion
from V1 to V2 at Temperature T
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CW03-05
Exercise 5.1 Isentropic Changes
Meteorologists often treat the bottom 10 km of the Earth’s atmosphere as isentropic (having
constant entropy). Assuming this to be true, and that the conditions at sea level are p = 1.000
atm and T = 300.0 K, what should the temperature be at an altitude of 10.0 km where the
pressure is 0.275 atm? Assume that air is an Ideal Gas with a constant volume heat capacity
of CV = 5R/2
Exercise 5.2 A Double Piston Apparatus Puzzle
A thermally insulated cylindrical container of fixed total volume is divided into three sections,
A, B, and C. Sections A and B are separated by a frictionless adiabatic piston (which transmits
work but no heat), and sections B and C are separated by a frictionless diabatic (thermally
conductive) piston. Each section of the cylinder contains 1.000 mol of an ideal gas with
CV = 5R/2. Initially, all three sections have a pressure of 1.000 atm and a temperature of
298.0 K. The gas is section A is heated (with an embedded resistive heater) slowly until the
temperature in section C reaches 348.0 K.
i What is the final Temperature (in K) of the gas in section B?
ii What is the final Volume (in L) in section B?
iii What is the final Volume (in L) in section A?
iv What is the final Temperature (in K) of the gas in section A?
Exercise 5.3 Heat Capacity along a Path
The heat capacity of an ideal gas is dependent on the path of the heating (or cooling) process.
A particular monoatomic (ideal) gas has a molar heat capacity of CV = 3R/2 when heated
(or cooled) at constant volume. This same gas has a molar heat capacity of Cp = 5R/2 when
heated at constant pressure.
i Why are the constant pressure and constant volume heat capacities different?
ii What is the molar heat capacity of this same gas when it is heated (reversibly) at constant
p/V , i.e. what is Cp/V ? (Constant p/V means that the ratio of the pressure to the volume of
the gas remains constant along the heating path)
— end of CW03 —
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