Chapter 10, Mathematical Methods Video Solutions, Heat and Thermodynamics

Problem 1

Starting with the first Maxwell relation, derive the remaining three by using only the relations:
$$
\left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1,
$$
and
$$
\left(\frac{\partial x}{\partial y}\right)_{f}\left(\frac{\partial y}{\partial z}\right)_{f}\left(\frac{\partial z}{\partial x}\right)_{f}=+1 .
$$

Nick Johnson

Numerade Educator

00:55

Problem 2

Show that, for an ideal gas:
(a) $A=\int C_{V} d T-T \int \frac{C_{V}}{T} d T-n R T \ln V-$ const. $T+$ const.
(b) $\quad G=\int C_{P} d T-T \int \frac{C_{P}}{T} d T+n R T \ln P-$ const. $T+$ const.
(c) Apply the above equations to $1 \mathrm{~mol}$ of an ideal gas.

Keshav Singh

Numerade Educator

01:53

Problem 3

From the differential equation for the thermodynamic potential $A(T, V)$, derive expressions for pressure $P$, entropy $S$, internal energy $U$, heat capacity at constant volume $C_{y}$, heat capacity at constant pressure $C_{P}$, volume expansivity $\beta$, and isothermal compressibility $\kappa$.

Dushyant Barot

Numerade Educator

01:57

Problem 4

Derive the following equations:
(a) $U=-T\left(\frac{\partial A}{\partial T}\right)_{V}=-T^{2}\left[\frac{\partial(A / T)}{\partial T}\right]_{V}$.
(b) $\quad C_{V}=-T\left(\frac{\partial^{2} A}{\partial T^{2}}\right)_{V}$.
(c) $\quad H=G-T\left(\frac{\partial G}{\partial T}\right)_{P}=-T^{2}\left[\frac{\partial(G / T)}{\partial T}\right]_{P} \quad$ (Gibbs-Helmholtz equation).
(d) $\quad C_{P}=-T\left(\frac{\partial^{2} G}{\partial T^{2}}\right)_{P}$.

Dan Ni

Numerade Educator

01:15

Problem 5

Another set of characteristic functions for a single-substance system can be defined by performing the Legendre transformations on the entropy $S(U, V)$ rather than on the internal energy $U(V, S)$. The thermodynamic potentials turn out to be particularly useful in statistical mechanics and the theory of irreversible thermodynamics, in contrast to equilibrium thermodynamics presented in this book.
(a) Show that Legendre transformation of $S(U, V)$ that produces the characteristic function $J(1 / T, V)$, known as the Massieu function, is given by the transform
$$
J=-\frac{U}{T}+S=-\frac{A}{T},
$$
and
$$
d J=\frac{U}{T^{2}} d T+\frac{P}{T} d V
$$
(b) Show that Legendre transformation of $J(1 / T, V)$ that produces the thermodynamic potential $Y(1 / T, P / T)$, known as the Planck function, is defined by the transform
and
$$
\begin{aligned}
&Y=-\frac{H}{T}+S=-\frac{G}{T} \\
&d Y=\frac{H}{T^{2}} d T-\frac{V}{T} d P
\end{aligned}
$$

James Kiss

Numerade Educator

03:17

Problem 6

From the fact that $d V / V$ is an exact differential, derive the relation
$$
\left(\frac{\partial \beta}{\partial P}\right)_{T}=-\left(\frac{\partial \kappa}{\partial T}\right)_{P}
$$

Narayan Hari

Numerade Educator

03:11

Problem 7

By invoking the condition for an exact differential, Eq. (10.30), demonstrate that the reversible heat $Q_{R}$ is not a thermodynamic property.

Mayank Tripathi

Numerade Educator

04:32

Problem 8

Derive the third $T^{\prime} d S$ equation,
$$
T d S=C_{V}\left(\frac{\partial T}{\partial P}\right)_{V} d P+C_{P}\left(\frac{\partial T}{\partial V}\right)_{F} d V_{+}
$$
and show that the three $T$ dS equations may be written as follows:
(a) $T d S=C_{V} d T+\frac{\beta T}{\kappa} d V$.
(b) $\quad T d S=C_{P} d T-V \beta T d P$
(c) $T d S=\frac{C_{V} \kappa}{\beta} d P+\frac{C_{P}}{\beta V} d V$.

Mahnoor Amin

Numerade Educator

02:16

Problem 9

The pressure on $500 \mathrm{~g}$ of copper is increased reversibly and isothermally from 0 to $5000 \mathrm{~atm}$ at $298 \mathrm{~K}$. (Take the density $\rho=8.96 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, volume expansivity $\beta=49.5 \times 10^{-6} \mathrm{~K}^{-1}$, isothermal compressibility $\kappa=6.18 \times 10^{-12} \mathrm{~Pa}^{-1}$, and specific
heat $c_{P}=385 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ to be constant.)
(a) How much heat is transferred during the compression?
(b) How much work is done during the compression?
(c) Determine the change of internal energy.
(d) What would have been the rise of temperature if the copper had been subjected to a reversible adiabatic compression?

Nick Johnson

Numerade Educator

07:39

Problem 10

The pressure on $0.2 \mathrm{~kg}$ of water is increased reversibly and isothermally from atmospheric pressure to $3 \times 10^{8} \mathrm{~Pa}$ at $20^{\circ} \mathrm{C}$. (Numerical values are given in Table 9.6.)
(a) How much heat is transferred?
(b) How much work is done?
(c) Calculate the change in internal energy.

Katie Mcalpine

Numerade Educator

01:05

Problem 11

The pressure on $1 \mathrm{~g}$ of water is increased from 0 to $10^{8} \mathrm{~Pa}$ reversibly and adiabatically. Calculate the temperature change when the initial temperature and other variables have the different values given in the three cases below:

Carson Merrill

Numerade Educator

02:37

Problem 12

A gas obeys the equation $P(v-b)=R T$, where $b$ is constant and $c_{V}$ is constant. Show that:
(a) $u$ is a function of $T$ only.
(b) $\gamma$ is constant.
(c) A relation that holds during an adiabatic process is
$$
P(\nu-b)^{\gamma}=\text { const. }
$$

Pritesh Ranjan

Numerade Educator

04:07

Problem 13

Show that for a gas obeying the van der Waals equation $\left(P+a / v^{2}\right)(v-b)=R T$, with $c_{V}$ a function of $T$ only, an equation for an adiabatic process is
$$
T(v-b)^{R / \epsilon r}=\text { const. }
$$

Mukesh Devi

Numerade Educator

04:59

Problem 14

(a) Using the virial expansion
$$
P \nu=R T\left(1+B P+C P^{2}+\cdots\right)
$$
calculate $(\partial u / \partial P)_{T}$ and its limit as $P \rightarrow 0$.
(b) Using the same expansion, calculate $(\partial P / \partial v)_{T}$ and its limit as $P \rightarrow 0$.
(c) Using parts $(a)$ and $(b)$, calculate $(\partial u / \partial v)_{T}$ and its limit as $P \rightarrow 0$. (Compare the solution with the results of Rossini and Frandsen given in Sec. $5.2$.)

Gaurav Gupta

Numerade Educator

04:32

Problem 15

Show that the differentials of the three thermodynamic potentials $U, H$, and $A$ may be written
$$
\begin{aligned}
&d U=\left(C_{P}-P V \beta\right) d T+V(\kappa P-\beta T) d P, \\
&d H=C_{P} d T+V(1-\beta T) d P
\end{aligned}
$$
and
$$
d A=-(P V \beta+S) d T+P V \kappa d P
$$

Mahnoor Amin

Numerade Educator

02:06

Problem 16

(a) Derive the equation
$$
\left(\frac{\partial C_{V}}{\partial V}\right)_{T}=T\left(\frac{\partial^{2} P}{\partial T^{2}}\right)_{V}
$$
(b) Prove that $C_{V}$ of an ideal gas is a function of $T$ only.
(c) In the case of a gas obeying the equation of state
$$
\frac{P v}{R T}=1+\frac{\beta}{v}
$$
where $B$ is a function of $T$ only, show that
$$
c_{V}=-\frac{R T}{v} \frac{d^{2}}{d T^{2}}(B T)+\left(c_{V}\right)_{0}
$$
where $\left(c_{V}\right)_{0}$ is the value at very large volumes.

Mukesh Devi

Numerade Educator

02:21

Problem 17

(a) Derive the equation
$$
\left(\frac{\partial C_{P}}{\partial P}\right)_{T}=-T\left(\frac{\partial^{2} V}{\partial T^{2}}\right)_{P}
$$
(b) prove that $C_{P}$ of an ideal gas is a function of $T$ only.
(c) In the case of a gas obeying the equation of state
$$
P v=R T+B P
$$
where $B$ is a function of $T$ only, show that
$$
c_{P}=-T \frac{d^{2} B}{d T^{2}} P+\left(c_{P}\right)_{0}
$$
where $\left(c_{P}\right)_{0}$ is the value at very low pressures.

Mukesh Devi

Numerade Educator

02:12

Problem 18

In the accompanying table are listed the thermal properties of liquid neon, compiled by Gladun. Calculate the plot against temperature: (a) $c_{V},(b) \kappa_{S}$, and $(c) \gamma$.

Pahk Thepchatri

Numerade Educator

01:57

Problem 19

Derive the following equations:
(a) $\quad C_{V}=-T\left(\frac{\partial P}{\partial T}\right)_{V}\left(\frac{\partial V}{\partial T}\right)_{S} .$
(b) $\left(\frac{\partial V}{\partial T}\right)_{S}=-\frac{C_{V} \kappa}{\beta T}$.
(c) $\frac{(\partial V / \partial T)_{S}}{(\partial V / \partial T)_{P}}=\frac{1}{1-\gamma}$.

Dan Ni

Numerade Educator

01:57

Problem 21

Derive the following equations:
(a) $\quad C_{P}=T\left(\frac{\partial V}{\partial T}\right)_{P}\left(\frac{\partial P}{\partial T}\right)_{S}$.
(b) $\left(\frac{\partial P}{\partial T}\right)_{S}=\frac{C_{P}}{V \beta T}$.
(c) $\frac{(\partial P / \partial T)_{S}}{(\partial P / \partial T)_{V}}=\frac{\gamma}{\gamma-1} .$

Dan Ni

Numerade Educator

02:28

Problem 21

(a) A measure of the result of an adiabatic Joule free expansion is provided by the Joule coefficient $\eta=(\partial T / \partial V)_{v}$. Show that
$$
\eta=-\frac{\mathrm{I}}{C_{V}}\left(\frac{\beta T}{\kappa}-P\right) .
$$
(b) A measure of the result of the Joule-Thomson expansion (adiabatic throttling process or isenthalpic expansion) is provided by the Joule-Thomson coefficient $\mu=(\partial T / \partial P)_{H}$. Show that
$$
\mu=\frac{V}{C_{P}}(\beta T-1)
$$

Manik Pulyani

Numerade Educator

02:11

Problem 22

The temperature of $1 \mathrm{~kg}$ of mercury at $20^{\circ} \mathrm{C}$ is increased by $5^{\circ} \mathrm{C}$ under conditions of constant volume. How much heating is required? (Take the volume expansivity $\beta=1.81 \times 10^{-4} \mathrm{~K}^{-1}$, specific heat at constant pressure $c_{P}=139 \mathrm{~J} / \mathrm{kg}+\mathrm{K}$, isothermal
compressibility $\kappa=3.94 \times 10^{-11} \mathrm{~Pa}^{-1}$ to be constant.)

Ajay Singhal

Numerade Educator