Exercises

1. Demonstrate that
   $$\left(\frac{\partial x}{\partial y}\right)_z=1\left/\left(\frac{\partial y}{\partial x}\right)_z\right..$$
   This result is known as the reciprocal rule of partial differentiation.

2. Prove Euler's chain rule:
   $$\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1.$$
   This result is also known as the cyclic rule of partial differentiation. Hence, deduce that
   $$\left(\frac{\partial x}{\partial y}\right)_z = -\left.\left(\frac... ...al z}{\partial y}\right)_x\right/\left(\frac{\partial z}{\partial x}\right)_y.$$

3. 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 $25^\circ$ 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.
   1. What is the increase in temperature of the water?
   2. How far from the left end of the cylinder will the piston come to rest?
   3. What is the increase in total entropy of the system?
   One atmosphere is equivalent to $10^{ 5} {\rm N m}^{-2}$ . Absolute zero is $-273^\circ$ C. The specific heat of water is $4.18$ joules/degree/gram. The density of water is 1 ${\rm g}/{\rm cm}^{ 3}$ .

4. A vertical cylinder contains $\nu$ moles of an ideal gas, and is closed off by a piston of mass $M$ and area $A$ . The acceleration due to gravity is $g$ . The molar specific heat $c_V$ (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 $V_0$ , and a temperature $T_0$ . The piston is now released, and, after some oscillations, comes to rest in a final equilibrium situation corresponding to a larger volume of gas.
   1. Does the temperature of the gas increase, decrease, or remain the same?
   2. Does the entropy of the gas increase, decrease, or remain the same?
   3. Show that the final temperature of the gas is
      $$T_1 = \left(\frac{c_V}{c_V + R}\right)T_0 + \left(\frac{M g/\nu A}{c_V + R}\right)V_0,$$
      where $R$ is the molar ideal gas constant.

5. The following describes a method used to measure the specific heat ratio, $\gamma \equiv c_p/c_V$ , of a gas. The gas, assumed ideal, is confined within a vertical cylindrical container, and supports a freely-moving piston of mass $m$ . The piston and cylinder both have the same cross-sectional area $A$ . Atmospheric pressure is $p_0$ , and when the piston is in equilibrium under the influence of gravity (acceleration $g$ ) and the gas pressure, the volume of the gas is $V_0$ . The piston is now displaced slightly from its equilibrium position, and is found to oscillate about this position at the angular frequency $\omega$ . 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
   $$\gamma = \frac{\omega^{ 2} m V_0}{m g A + p_0 A^{ 2}}.$$

6. 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 $dx$ , show that the pressure, $p(x,t)$ , in the fluid depends on the position, $x$ , and the time, $t$ , so as to satisfy the wave equation

   $$\frac{\partial^{ 2} p}{\partial t^{ 2}} = u^{ 2} \frac{\partial^{ 2} p}{\partial x^{ 2}}$$
   where the velocity of sound propagation, $u$ , is a constant given by $u=(\rho \kappa_S)^{-1/2}$ . Here $\rho$ is the equilibrium mass density of the fluid, and $\kappa_S$ is its adiabatic compressibility,
   $$\kappa_S= - \frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_S:$$
   that is, its compressibility measured under conditions in which the fluid is thermally insulated.

7. Demonstrate that
   $$\kappa_S = \frac{c_V}{c_p} \kappa_T,$$
   where $\kappa_T$ is the isothermal compressibility. Hence, deduce that
   $$\kappa_T-\kappa_S = \frac{v T \alpha_V^{ 2}}{c_p},$$
   where $v$ is the molar volume, and $\alpha_V$ the volume coefficient of expansion.

8. Refer to the results of the preceding two problems.
   1. Show that adiabatic compressibility, $\kappa_S$ , of an ideal gas is
      $$\kappa_S= \frac{1}{\gamma p},$$
      where $\gamma$ is the ratio of specific heats, and $p$ the gas pressure.
   2. Show that the velocity of sound in an ideal gas is
      $$u = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{ \gamma R T}{\mu}},$$
      where $\rho$ is the mass density, $R$ the molar ideal gas constant, $T$ the absolute temperature, and $\mu$ the molecular weight.
   3. How does the sound velocity depend on the gas temperature, $T$ , at a fixed pressure? How does it depend on the gas pressure, $p$ , at fixed temperature?
   4. Calculate the velocity of sound in nitrogen (${\rm N}_2$ ) gas at room temperature and pressure (i.e., $15^\circ$ C at 1 bar). Take $\gamma=1.4$ .

9. Show that, in general,
   $$\left(\frac{\partial\alpha_V}{\partial p}\right)_T+\left(\frac{\partial\kappa_T}{\partial T}\right)_p=0,$$
   where $\alpha_V$ is the volume coefficient of expansion, and $\kappa_T$ the isothermal compressibility.

10. Show that
    $$\left(\frac{\partial E}{\partial T}\right)_p = C_p - p V \alpha_V,$$
    and
    $$\left(\frac{\partial E}{\partial p}\right)_T =p V \kappa_T - (C_p-C_V) \frac{\kappa_T}{\alpha_V}.$$

11. Liquid mercury at $0^\circ$ C (i.e., 273K) has a molar volume $v=1.472\times 10^{-5} {\rm m}^{ 3}$ , a molar specific heat at constant pressure $c_p=28.0 {\rm J} {\rm mol}^{-1}$ , a volume coefficient of expansion $\alpha_V = 1.81\times 10^{-4} {\rm K}^{-1}$ , and an isothermal compressibility $\kappa_T = 3.88\times 10^{-11} ({\rm N m}^{-2})^{-1}$ . Find its molar specific heat at constant volume, and the ratio $\gamma=c_p/c_V$ .

12. Starting from the first Maxwell relation,
    $$\left(\frac{\partial T}{\partial V}\right)_S=-\left(\frac{\partial p}{\partial S}\right)_V,$$
    derive the other three by making use of the reciprocal and cyclic rules of partial differentiation (see Exercises 1 and 2), as well as the identity
    $$\left(\frac{\partial x}{\partial y}\right)_f\left(\frac{\partial y}{\partial z}\right)_f\left(\frac{\partial z}{\partial x}\right)_f=1.$$

13. Consider a van der Waals gas whose equation of state is
    $$\left(p+\frac{a}{v^{ 2}}\right)(v-b) = R T.$$
    The critical point is defined as the unique point at which $(\partial^{ 2} p/\partial v^{ 2})_T=(\partial p/\partial v)_T = 0$ . (See Section 9.10.) Let $p_c$ , $v_c$ , and $T_c$ be the temperature, molar volume, and temperature, respectively, at the critical point. Demonstrate that $p_c=a/(27 b^{ 2})$ , $v_c=3 b$ , and $T_c = 8 a/(27 R b)$ . Hence, deduce that the van der Waals equation of state can be written in the reduced form
    $$\left(p'+\frac{3}{v'^{ 2}}\right)\left(v'-\frac{1}{3}\right)=\frac{8 T'}{3},$$
    where $p'=p/p_c$ , $v'=v/v_c$ , and $T'=T/T_c$ .

    Figure 6.4: $p$ -$V$ diagram for the Otto cycle.

    

14. 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:

    $e\rightarrow a$ : Isobaric (i.e., constant pressure) intake (at atmospheric pressure).

    $a\rightarrow b$ : Adiabatic compression (compression stroke).

    $b\rightarrow c$ : 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.)

    $c\rightarrow d$ : Adiabatic expansion (power stroke).

    $d\rightarrow a$ : Isochoric decrease of temperature (exhaust value opened).

    $a\rightarrow e$ : Isobaric exhaust (at atmospheric pressure).

    1. Assume that the working substance is an ideal gas. Show that the efficiency of the cycle is
       $$\eta = 1-\left(\frac{T_d-T_a}{T_c-T_b}\right)=1-\frac{1}{r^{ \gamma-1}},$$
       where $r=V_1/V_2$ is the compression ratio of the engine.
    2. Calculate $\eta$ for the realistic values $r=5$ and $\gamma=1.5$ .
    Note: Because $\gamma>1$ , maximizing $r$ maximizes the engine efficiency. The maximum practical value of $r$ is about 7. For greater values, the rise in temperature during compression is large enough to cause ignition prior to the advent of the spark. This process is called 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.