- Demonstrate that
*reciprocal rule*of partial differentiation. - Prove
*Euler's chain rule*:*cyclic rule*of partial differentiation. Hence, deduce that - A cylindrical container 80 cm long is separated into two compartments by a
thin piston, originally clamped in position 30 cm from the left end. The left
compartment is filled with one mole of helium gas at a pressure of 5
atmospheres; the right compartment is filled with argon gas at 1 atmosphere of
pressure. The gases may be considered ideal. The cylinder is submerged in
1 liter of water, and the entire system is initially at the uniform temperature
of
C (with the previously mentioned pressures in the two compartments).
The heat capacities of the cylinder and piston may be neglected. When the piston
is unclamped, a new equilibrium situation is ultimately reached with the
piston in a new position.
- What is the increase in temperature of the water?
- How far from the left end of the cylinder will the piston come to rest?
- What is the increase in total entropy of the system?

- A vertical cylinder contains
moles of an ideal gas, and is closed off
by a piston of mass
and area
. The acceleration due to gravity is
.
The molar specific heat
(at constant volume) of the gas is a constant
independent of temperature. The heat capacities of the piston and cylinder
are negligibly small, and any frictional forces between the piston and
the cylinder walls can be neglected. The pressure of the atmosphere can also be neglected. The whole system is thermally insulated.
Initially, the piston is clamped in position so that the gas has a volume
,
and a temperature
. The piston is now released, and, after some oscillations,
comes to rest in a final equilibrium situation corresponding to a larger volume of
gas.
- Does the temperature of the gas increase, decrease, or remain the same?
- Does the entropy of the gas increase, decrease, or remain the same?
- Show that the final temperature of the gas is

- The following describes a method used to measure the specific heat ratio,
, of a gas. The gas, assumed ideal, is confined within a
vertical cylindrical container, and supports a freely-moving piston of mass
.
The piston and cylinder both have the same cross-sectional area
. Atmospheric
pressure is
, and when the piston is in equilibrium under the influence
of gravity (acceleration
) and the gas pressure, the volume of the gas is
.
The piston is now displaced slightly from its equilibrium position, and is found
to oscillate about this position at the angular frequency
. The oscillations of
the piston are sufficiently slow that the gas always remains in internal equilibrium,
but fast enough that the gas cannot exchange heat with its environment. The variations
in gas pressure and volume are therefore adiabatic.
Show that
- When sound passes through a fluid (liquid or gas), the period of vibration
is short compared to the relaxation time necessary for a macroscopically small
element of the fluid to exchange energy with the rest of the fluid through heat
flow. Hence, compressions of such an element of volume can be considered
adiabatic.
By analyzing one-dimensional compressions and rarefactions of the system of fluid contained in a slab of thickness , show that the pressure, , in the fluid depends on the position, , and the time, , so as to satisfy the wave equation

*adiabatic compressibility*, - Demonstrate that
- Refer to the results of the preceding two problems.
- Show that adiabatic compressibility,
, of an ideal gas is
- Show that the velocity of sound in an ideal gas
is
- How does the sound velocity depend on the gas temperature, , at a fixed pressure? How does it depend on the gas pressure, , at fixed temperature?
- Calculate the velocity of sound in nitrogen ( ) gas at room temperature and pressure (i.e., C at 1 bar). Take .

- Show that adiabatic compressibility,
, of an ideal gas is
- Show that, in general,
- Show that
- Liquid mercury at
C (i.e., 273K) has a
molar volume
, a molar specific heat at constant pressure
, a
volume coefficient of expansion
, and an isothermal
compressibility
. Find its molar specific heat at constant volume,
and the ratio
.
- Starting from the first Maxwell relation,
- Consider a van der Waals gas whose equation of state is
- The behavior of a four-stroke gasoline engine can be approximated by the so-called
*Otto cycle*, shown in Figure 6.4. The cycle is as follows:- : Isobaric (i.e., constant pressure) intake (at atmospheric pressure).
- : Adiabatic compression (compression stroke).
- : Isochoric (i.e., constant volume) increase of temperature during ignition. (Gas combustion is an irreversible process. Here, it is replaced by a reversible isochoric process in which heat is assumed to flow into the system from a reservoir.)
- : Adiabatic expansion (power stroke).
- : Isochoric decrease of temperature (exhaust value opened).
- : Isobaric exhaust (at atmospheric pressure).

- Assume that the working substance is an ideal gas. Show that the efficiency of the cycle is
- Calculate for the realistic values and .

*pre-ignition*, and is deleterious to the operation of the engine. Pre-ignition is not a problem for diesel engines (which depend on spontaneous ignition, rather than triggering ignition via a spark), so higher compression ratios are possible. This is partly the reason that diesel engines are inherently more efficient than gasoline engines.