Chapter 15, Chemical Equilibrium Video Solutions, Heat and Thermodynamics

Problem 1

Prove Gibbs' theorem by using the apparatus depicted in Fig. $15-2$ in such a way that the gases are separated reversibly and adiabatically.

Brooke Smith

Numerade Educator

01:03

What is the minimum amount of work required to separate 1 mol of air at $27^{\circ} \mathrm{C}$ and 1 atm pressure (assumed to be composed of $\frac{1}{5} \mathrm{O}_{2}$ and $\frac{4}{3} \mathrm{~N}_{2}$ ) into $\mathrm{O}_{2}$ and $\mathrm{N}_{2}$, each at $27^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ pressure?

Narayan Hari

Numerade Educator

01:32

Problem 3

Calculate the entropy change of the universe due to the diffusion of two ideal gases $(1 \mathrm{~mol}$ of each $)$ at the same temperature and pressure, by calculating $\int \mathrm{d} Q / T$ over a series of reversible processes involving the use of the apparatus depicted in Fig. $15-2 .$

Manik Pulyani

Numerade Educator

02:05

Problem 4

There are $n_{1}$ moles of an ideal monatomic gas at temperature $T_{1}$ and pressure $P$ in one compartment of an insulated container. In an adjoining compartment separated by an insulating partition are $n_{2}$ moles of another ideal monatomic gas at temperature $T_{2}$ and pressure $P .$ When the partition is removed:
(a) Show that the final pressure of the mixture is $P$.
(b) Calculate the entropy change when the gases are identical.
(c) Calculate the entropy change when the gases are different.

Penny Riley

Numerade Educator

02:05

Problem 5

There are $n_{1}$ moles of an ideal gas at pressure $P_{1}$ and temperature $T$ in one compartment of an insulated container. In an adjoining compartment separated by a partition are $n_{2}$ moles of an ideal gas at pressure $P_{2}$ and temperature $T .$ When the partition is removed:
(a) Calculate the final pressure of the mixture.
(b) Calculate the entropy change when the gases are identical.
(c) Calculate the entropy change when the gases are different.
(d) Prove that the entropy change in part $(c)$ is the same as that which would be produced by two independent free expansions.

Penny Riley

Numerade Educator

01:27

Problem 6

Suppose that we have $1 \mathrm{~mol}$ of a monatomic gas $B_{1}$, whose nuclei are in their lowest energy state, and $1 \mathrm{~mol}$ of a monatomic gas $B_{2}$ consisting of exactly the same atoms as $B_{1}$, except that the nuclei are in an excited state whose energy $\epsilon$ is much larger than $k T$ and whose lifetime is much larger than the time for diffusion to occur. Both gases are at the same pressure and are maintained at the same temperature $T$ by a heat reservoir.
(a) Immediately after the nuclei of gas $B_{2}$ have been excited, diffusion takes place. Calculate the entropy change of the universe.
(b) After the nuclei of gas $B_{2}$ have been excited, a time much larger than the lifetime of the excited state is allowed to elapse, and then diffusion takes place. Calculate the entropy change of the universe.
(c) Show that the answer to part $(b)$ is larger than that to part $(a)$. (This problem is due to M. J. Klein.)

Manik Pulyani

Numerade Educator

13:45

Problem 7

Consider the system depicted in Fig. $\mathrm{P} 15-1$, where the whole system and also each half are in equilibrium. Consider a process to take place in which each half of the system remains at the constant volume $V / 2$, but a small amount of heat $\delta U$ (at
constant volume, $d Q=d U$ ) is extracted from the left half and transferred to the right half, as shown in Fig. P15-1. Realizing that
$$
\left(\frac{\partial S_{L}}{\partial U}\right)_{V}=\left(\frac{\partial S}{\partial U}\right)_{V}
$$
(a) Expand the entropy $S_{L}$ of the left half by means of a Taylor series about the equilibrium value $S_{\max } / 2$, terminating the series after the squared term. Do the same for $S_{R}$.
(b) Show that
$$
\delta S_{U, V}=S_{L}+S_{R}-S_{\max }=\left(\frac{\partial^{2} S}{\partial U^{2}}\right)_{V}(\delta U)^{2} .
$$
(c) Show that $C_{V}>0$, which is the condition for thermal stability.

Jack Hou

Numerade Educator

00:58

Problem 8

(a) Show that the molar Helmholtz function $a$ of an ideal gas is
$$
a=u_{0}-T \int \frac{\int c_{V} d T}{T^{2}} d T-T s_{0}-R T \ln v
$$
(b) Show that the Helmholtz function of a mixture of inert ideal gases is
$$
A=\sum n_{j}\left(a_{j}+R T \ln x_{j}\right)
$$
(c) Show that the change in the Helmholtz function due to diffusion is
$$
A_{f}-A_{i}=R T \sum n_{j} \ln x_{j}
$$

Hast Aggarwal

Numerade Educator

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Problem 9

Consider a hydrostatic system of constant mass maintained in thermal and mechanical equilibrium. It is not in chemical equilibrium, however, because of chemical reactions and transport of matter between phases. Under these circumstances, the system undergoes an irreversible cycle. Prove that
$$
\oint \frac{\mathrm{d} Q}{T}<0
$$

Victor Salazar

Numerade Educator

10:06

Problem 10

By means of the equations $d G=-S d T+V d P+\sum \mu_{j} d n_{j}$ and $G=\sum \mu_{j} n_{j}$, prove that:
(a) $-S d T+V d P=\sum n_{j} d \mu_{j}$.
(b) $-s d T+v d P=\sum x_{j} d \mu_{i}$

Chris Trentman

Numerade Educator

02:15

Problem 11

Show that, if the internal energy of a phase is expressed as a function of $S, V, n_{1}$, $n_{2} \ldots, n_{c}$, then:
(a) $d U=T d S-P d V+\sum \mu_{j} d n_{j}, \quad$ where $\mu_{j}=\left(\partial U / \partial n_{j}\right)_{S, V, \text { other } r^{\prime} s^{\circ}}$
(b) $U=T S-P V+\sum \mu_{j} n_{j}$
(c) $-S d T+V d P=\sum n_{j} d \mu_{j}$

Sheh Lit Chang

University of Washington

03:53

Problem 12

Show that, if the Helmholtz function of a phase is expressed as a function of $T, V, n_{1}, n_{2}, \ldots, n_{c}$, then:
(a) $d A=-S d T-P d V+\sum \mu_{j} d n_{j}, \quad$ where $\mu_{j}=\left(\partial A / \partial n_{j}\right)_{T, V, \text { other } n^{\prime} s^{\circ}}$
(b) $A=-P V+\sum \mu_{j} n_{j}$
(c) $-S d T+V d P=\sum n_{j} d \mu_{j}$

Foster Wisusik

Numerade Educator

01:08

Problem 13

Prove that:
(a) $T d\left(\frac{A}{T}\right)=-\frac{U}{T} d T-P d V+\sum \mu_{j} d n_{j}$
(b) $d(P V)=S d T+V d P+\sum n_{j} d \mu_{j}$

Yujie Wang

College of San Mateo

00:55

Problem 14

Show that, for an ideal gas in a mixture of ideal gases,
$$
d \mu_{j}=\frac{\mu_{j}-h_{j}}{T} d T+v_{i} d P+R T d \ln x_{j}
$$

Keshav Singh

Numerade Educator

02:05

Problem 15

Consider a uniform substance of variable mass. Suppose that, at any moment, there are $n$ moles of substance and that the system undergoes an infinitesimal reversible process in which $n$ changes by $d n$. For any 1 mol of substance,
$$
\mathrm{d} q=T d s=d u+P d \nu
$$
and the total heat transfer $d Q=n$ ? $q=n T d s$.
(a) Prove that
$$
\mathrm{d} Q=d U+P d V-h d n
$$
where $U$ and $V$ refer to the entire system, but $h$ is the molar enthalpy.
(b) From Prob. $15.11(a)$ show that
$$
T d S=d U+P d V-g d n
$$
(c) How does $T$ dS differ from dQ?
(d) Show that, if $T$ and $P$ remain constant, then ? $Q=0$ but $T d S$ does not.

James Irizarry

Numerade Educator

04:57

Problem 16

Starting with $n_{0}$ moles of $\mathrm{CO}$ and $n_{0}$ moles of $\mathrm{H}_{2} \mathrm{O}$ capable of undergoing the reaction
$$
\mathrm{CO}+\mathrm{H}_{2} \mathrm{O}=\mathrm{CO}_{2}+\mathrm{H}_{2}
$$
in the gaseous phase, set up a table of values of $A, v, n$, and $x$, similar to that of Table 15.1.

Bryan Li

Numerade Educator

05:05

Problem 17

Starting with $n_{0}$ moles of $\mathrm{H}_{2} \mathrm{~S}$ and $2 n_{0}$ moles of $\mathrm{H}_{2} \mathrm{O}$, capable of undergoing the reaction
$$
\mathrm{H}_{2} \mathrm{~S}+2 \mathrm{H}_{2} \mathrm{O} \rightleftharpoons 3 \mathrm{H}_{2}+\mathrm{SO}_{2}
$$
in the gaseous phase, set up a table of values of $A, v, n$, and $x$, similar to that of Table $15.1$.

Osman Elomda

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

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Problem 7

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Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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Problem 15

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Problem 17