15.9: Appendix I- Answers to Selected Problems

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3.3(b)
\(q = -w = 1.00\timesten{5}\units{J}\)

3.4(c)
\(w=1.99\timesten{3}\units{J}\), \(q=-1.99\timesten{3}\units{J}\).

3.5
\(0.0079\%\)

3.6(c)
\(V_2 \ra nRV_1/(C_V+nR)\), \(T_2 \ra \infty\). For \(C_V=(3/2)nR\), \(V_2/V_1 \ra 0.4\).

3.11
\(9.58\timesten{3}\units{s}\) (\(2\units{hr}\) \(40\units{min}\))

4.4
\(\Del S=0.054\units{J K\(^{-1}\)}\)

4.5
\(\Del S = 549\units{J K\(^{-1}\)}\) for both processes; \(\int\!\dq/T\subs{ext} = 333\units{J K\(^{-1}\)}\) and \(0\).

5.4(a)
\( \D S = nR\ln\left[cT^{3/2}\left(\frac{V}{n}-b\right)\right] + \left(\frac{5}{2}\right)nR \)

5.5(a)
\(q=0\), \(w=1.50\timesten{4}\units{J}\), \(\Del U=1.50\timesten{4}\units{J}\), \(\Del H=2.00\timesten{4}\units{J}\)

5.5(c)
\(\Del S=66.7\units{J K\(^{-1}\)}\)

6.1
\(S\m \approx 151.6\units{J K\(^{-1}\) mol\(^{-1}\)}\)

7.6(a)
\(\alpha=8.519\timesten{-4}\units{K\(^{-1}\)}\)
\(\kappa_t=4.671\timesten{-5}\units{bar\(^{-1}\)}\)
\(\pd{p}{T}{V}=18.24\units{bar K\(^{-1}\)}\)
\(\pd{U}{V}{T}=5437\br\)

7.6(b)
\(\Del p \approx 1.8\br\)

7.7(b)
\(\pd{\Cpm}{p}{T}=-4.210\timesten{-8}\units{J K\(^{-1}\) Pa\(^{-1}\) mol\(^{-1}\)}\)

7.8(b)
\(8\timesten{-4}\units{K\(^{-1}\)}\)

7.11
\(5.001\timesten{3}\units{J}\)

7.12
\(\Del H = 2.27\timesten{4}\units{J}\), \(\Del S = 43.6\units{J K\(^{-1}\)}\)

7.13(a)
\(\Cpm\st=42.3\units{J K\(^{-1}\) mol\(^{-1}\)}\)

7.13(b)
\(\Cpm \approx 52.0\units{J K\(^{-1}\) mol\(^{-1}\)}\)

7.14(a)
\(2.56\units{J K\(^{-1}\) g\(^{-1}\)}\)

7.15(b)
\(\fug = 17.4\br\)

7.16(a)
\(\phi=0.739\), \(\fug=148\br\)

7.16(b)
\(B = -7.28\timesten{-5}\units{m\(^3\) mol\(^{-1}\)}\)

8.2(a)
\(S\m\st\liquid =253.6\units{J K\(^{-1}\) mol\(^{-1}\)}\)

8.2(b)
\(\Delsub{vap}S\st=88.6\units{J K\(^{-1}\) mol\(^{-1}\)}\), \(\Delsub{vap}H\st=2.748\timesten{4}\units{J mol\(^{-1}\)}\)

8.4
\(4.5\timesten{-3}\br\)

8.5
\(19\units{J mol\(^{-1}\)}\)

8.6(a)
\(352.82\K\)

8.6(b)
\(3.4154\timesten{4}\units{J mol\(^{-1}\)}\)

8.7(a)
\(3.62\timesten{3}\units{Pa K\(^{-1}\)}\)

8.7(b)
\(3.56\timesten{3}\units{Pa K\(^{-1}\)}\)

8.7(c)
\(99.60\units{\(\degC\)}\)

8.8(b)
\(\Delsub{vap}H\st = 4.084\timesten{4}\units{J mol\(^{-1}\)}\)

8.9
\(0.93\units{mol}\)

9.2(b)
\(V\A(x\B=0.5) \approx 125.13\units{cm\(^3\) mol\(^{-1}\)}\)
\(V\B(x\B=0.5)\approx 158.01\units{cm\(^3\) mol\(^{-1}\)}\)
\(V\B^{\infty} \approx 157.15\units{cm\(^3\) mol\(^{-1}\)}\)

9.4
real gas: \(p=1.9743\br\)
ideal gas: \(p=1.9832\br\)

9.5(a)
\(x\subs{N\(_2\)} = 8.83\timesten{-6}\)
\(x\subs{O\(_2\)} = 4.65\timesten{-6}\)
\(y\subs{N\(_2\)} = 0.763\)
\(y\subs{O\(_2\)} = 0.205\)

9.5(b)
\(x\subs{N\(_2\)} = 9.85\timesten{-6}\)
\(x\subs{O\(_2\)} = 2.65\timesten{-6}\)
\(y\subs{N\(_2\)} = 0.851\)
\(y\subs{O\(_2\)} = 0.117\)

9.7(b)
\(\fug\A=0.03167\br\), \(\fug\A=0.03040\br\)

9.8(a)
In the mixture of composition \(x\A=0.9782\), the activity coefficient is \(\g\B \approx 11.5\).

9.9(d)
\(k\subs{H,A} \approx 680\units{kPa}\)

9.11
Values for \(m\B/m\st=20\): \(\g\A=1.026\), \(\g\mbB=0.526\); the limiting slopes are \(\dif\g\A/\dif(m\B/m\st)=0\), \(\dif\g\mbB/\dif(m\B/m\st)=-0.09\)

9.13
\(p\subs{N\(_2\)} = 0.235\br\)
\(y\subs{N\(_2\)} = 0.815\)
\(p\subs{O\(_2\)} = 0.0532\br\)
\(y\subs{O\(_2\)} = 0.185\)
\(p=0.288\br\)

9.14(b)
\(h=1.2\units{m}\)

9.15(a)
\(p(7.20\units{cm})-p(6.95\units{cm})=1.2\br\)

9.15(b)
\(M\B=187\units{kg mol\(^{-1}\)}\)
mass binding ratio \({} = 1.37\)

10.2
\( \g{\pm} = 0.392 \)

11.1
\(\Delsub{r}H\st = -63.94\units{kJ mol\(^{-1}\)}\)
\(K=4.41\timesten{-2}\)

11.2(b)
\(\Delsub{f}H\st\): no change
\(\Delsub{f}S\st\): subtract \(0.219\units{J K\(^{-1}\) mol\(^{-1}\)}\)
\(\Delsub{f}G\st\): add \(65\units{J mol\(^{-1}\)}\)

11.3
\(p(298.15\K)=2.6\timesten{-6}\br\)
\(p(273.15\K) = 2.7\timesten{-7}\br\)

11.4(a)
\(-240.34\units{kJ mol\(^{-1}\)}\), \(-470.36\units{kJ mol\(^{-1}\)}\), \(-230.02\units{kJ mol\(^{-1}\)}\)

11.4(b)
\(-465.43\units{kJ mol\(^{-1}\)}\)

11.4(c)
\(-39.82\units{kJ mol\(^{-1}\)}\)

11.5
\(\Del H = 0.92\units{kJ}\)

11.6
\(L\A=-0.405\units{J mol\(^{-1}\)}\)
\(L\B=0.810\units{kJ mol\(^{-1}\)}\)

11.7(a)
State 1:
\(n\subs{C\(_6\)H\(_{14}\)}=7.822\timesten{-3}\mol\)
\(n\subs{H\(_2\)O}=0.05560\mol\)
amount of O\(_2\) consumed: \(0.07431\mol\)
State 2:
\(n\subs{H\(_2\)O}=0.11035\mol\)
\(n\subs{CO\(_2\)}=0.04693\mol\)
\tx{mass of H\(_2\)O}=\(1.9880\units{g}\)

11.7(b)
\(V\m\tx{(C\(_6\)H\(_{14}\))} = 131.61\units{cm\(^3\) mol\(^{-1}\)}\)
\(V\m\tx{(H\(_2\)O)} = 18.070\units{cm\(^3\) mol\(^{-1}\)}\)

11.7(c)
State 1: \(V\tx{(C\(_6\)H\(_{14}\))} = 1.029\units{cm\(^3\)}\)
\(V\tx{(H\(_2\)O)} = 1.005\units{cm\(^3\)}\)
\(V\sups{g} = 348.0\units{cm\(^3\)}\)
State 2:
\(V\tx{(H\(_2\)O)} = 1.994\units{cm\(^3\)}\)
\(V\sups{g} = 348.0\units{cm\(^3\)}\)

11.7(d)
State 1:
\(n\subs{O\(_2\)}=0.429\mol\)
State 2:
\(n\subs{O\(_2\)}=0.355\mol\)
\(y\subs{O\(_2\)}=0.883\)
\(y\subs{CO\(_2\)}=0.117\)

11.7(e)
State 2:
\(p_2 = 27.9\br\)
\(p\subs{O\(_2\)} = 24.6\br\)
\(p\subs{CO\(_2\)} = 3.26\br\)

11.7(f)
\(\fug\subs{H\(_2\)O}(0.03169\br ) = 0.03164\br\)
State 1: \(\fug\subs{H\(_2\)O} = 0.03234\br\)
State 2: \(\fug\subs{H\(_2\)O} = 0.03229\br\)

11.7(g)
State 1:
\(\phi\subs{H\(_2\)O}=0.925\)
\(\phi\subs{O\(_2\)}=0.981\)
\(\fug\subs{O\(_2\)}= 29.4\br\)
State 2:
\(\phi\subs{H\(_2\)O}=0.896\)
\(\phi\subs{O\(_2\)}=0.983\)
\(\phi\subs{CO\(_2\)}=0.910\)
\(\fug\subs{O\(_2\)}=24.2\br\)
\(\fug\subs{CO\(_2\)}=2.97\br\)

11.7(h)
State 1:
\(n\subs{H\(_2\)O}\sups{g}=5.00\timesten{-4}\mol\)
\(n\subs{H\(_2\)O}\sups{l} =0.05510\mol\)
State 2:
\(n\subs{H\(_2\)O}\sups{g}=5.19\timesten{-4}\mol\)
\(n\subs{H\(_2\)O}\sups{l} =0.10983\mol\)

11.7(i)
State 1:
\(k_{m,\tx{O\(_2\)}}= 825\units{bar kg mol\(^{-1}\)}\)
\(n\subs{O\(_2\)} = 3.57\timesten{-5}\mol\)
State 2:
\(k_{m,\tx{O\(_2\)}}= 823\units{bar kg mol\(^{-1}\)}\)
\(k_{m,\tx{CO\(_2\)}}= 30.8\units{bar kg mol\(^{-1}\)}\)
\(n\subs{O\(_2\)} = 5.85\timesten{-5}\mol\)
\(n\subs{CO\(_2\)} = 1.92\timesten{-4}\mol\)

11.7(j)
H\(_2\)O vaporization: \(\Del U = +20.8\units{J}\)
H\(_2\)O condensation: \(\Del U = -21.6\units{J}\)

11.7(k)
O\(_2\) dissolution: \(\Del U = -0.35\units{J}\)
O\(_2\) desolution: \(\Del U = 0.57\units{J}\)
CO\(_2\) desolution: \(\Del U = 3.32\units{J}\)

11.7(l)
C\(_6\)H\(_{14}\)(l) compression: \(\Del U=-1.226\units{J}\)
solution compression: \(\Del U=-0.225\units{J}\)
solution decompression: \(\Del U=0.414\units{J}\)

11.7(m)
O\(_2\) compression: \(\Del U=-81\units{J}\)
gas mixture: \(\dif B/\dif T = 0.26\timesten{-6}\units{m\(^3\)K\(^{-1}\) mol\(^{-1}\)}\)
gas mixture expansion: \(\Del U=87\units{J}\)

11.7(n)
\(\Del U = 8\units{J}\)

11.7(o)
\(\Delsub{c}U\st = -4154.4\units{kJ mol\(^{-1}\)}\)

11.7(p)
\(\Delsub{c}H\st = -4163.1\units{kJ mol\(^{-1}\)}\)

11.8
\(\Delsub{f}H\st = -198.8\units{kJ mol\(^{-1}\)}\)

11.9
\(T_2=2272\K\)

11.10
\(p\tx{(O\(_2\))} =2.55\timesten{-5}\br\)

11.11(a)
\(K=3.5\timesten{41}\)

11.11(b)
\(p\subs{H\(_2\)}=2.8\timesten{-42}\br\)
\(N\subs{H\(_2\)}=6.9\timesten{-17}\)

11.11(c)
\(t=22\units{s}\)

11.12(b)
\(p \approx 1.5\timesten{4}\br\)

11.13(c)
\(K=0.15\)

12.1(b)
\(T=1168\K\)
\(\Delsub{r}H\st=1.64\timesten{5}\units{J mol\(^{-1}\)}\)

12.4
\(K\subs{f}=1.860\units{K kg mol\(^{-1}\)}\)
\(K\bd=0.5118\units{K kg mol\(^{-1}\)}\)

12.5
\(M\B \approx 5.6\timesten{4}\units{g mol\(^{-1}\)}\)

12.6
\(\Delsub{sol,B}H\st/\tx{kJ mol\(^{-1}\)}=-3.06, 0, 6.35\)
\(\Delsub{sol,B}S\st/\tx{J K\(^{-1}\) mol\(^{-1}\)} = -121.0, -110.2, -88.4\)

12.7(a)
\( m_+\aph = m_-\aph = 1.20\timesten{-3}\units{mol kg\(^{-1}\)} \)
\( m_+\bph = 1.80\timesten{-3}\units{mol kg\(^{-1}\)} \)
\( m_-\bph = 0.80\timesten{-3}\units{mol kg\(^{-1}\)} \)
\( m\subs{P} = 2.00\timesten{-6}\units{mol kg\(^{-1}\)} \)

12.8(a)
\(p\sups{l} =2.44\br\)

12.8(b)
\(\fug(2.44\br)-\fug(1.00\br)=3.4\timesten{-5}\br\)

12.10(a)
\(x\B=1.8\timesten{-7}\)
\(m\B = 1.0\timesten{-5}\units{mol kg\(^{-1}\)}\)

12.10(b)
\(\Delsub{sol,B}H\st = -1.99\timesten{4}\units{J mol\(^{-1}\)}\)

12.10(c)
\(K=4.4\timesten{-7}\)
\(\Delsub{r}H\st=9.3\units{kJ mol\(^{-1}\)}\)

12.13(a)
\(p=92399.6\Pa\), \(y\B=0.965724\)

12.13(b)
\(\phi\A=0.995801\)

12.13(c)
\(\fug\A = 3164.47\Pa\)

12.13(d)
\(y\B = 0.965608\)

12.13(e)
\(Z = 0.999319\)

12.13(f)
\(p = 92347.7\Pa\)

12.13(g)
\(\kHB = 4.40890\timesten{9}\Pa\)

12.15(a)
\(\g\xbB=0.9826\)

12.15(b)
\(x\B=4.19\timesten{-4}\)

12.16
\(K=1.2\timesten{-6}\)

12.17(a)
\(\alpha=0.129\)
\(m_+=1.29\timesten{-3}\units{mol kg\(^{-1}\)}\)

12.17(b)
\(\alpha=0.140\)

12.18
\(\Delsub{f}H\st(\tx{Cl\(^-\), aq}) = -167.15\units{kJ mol\(^{-1}\)}\)
\(S\m\st(\tx{Cl\(^-\), aq}) = 56.46\units{J K\(^{-1}\) mol\(^{-1}\)}\)

12.19(a)
\(K\subs{s} = 1.783\timesten{-10}\)

12.20(a)
\(\Delsub{r}H\st=-65.769\units{kJ mol\(^{-1}\)}\)

12.20(b)
\(\Delsub{f}H\st\tx{(Ag\(^+\), aq)} =105.84\units{kJ mol\(^{-1}\)}\)

13.1(a)
\(F=4\)

13.1(b)
\(F=3\)

13.1(c)
\(F=2\)

13.10(a)
\(x\B\tx{(top)} =0.02\), \(x\B\tx{(bottom)} =0.31\)

13.10(b)
\(n\A = 2.1\mol\), \(n\B = 1.0 \mol\)

14.3(a)
\(\Delsub{r}G\st=-21.436\units{kJ mol\(^{-1}\)}\)
\(\Delsub{r}S\st=-62.35\units{J K\(^{-1}\) mol\(^{-1}\)}\)
\(\Delsub{r}H\st=-40.03\units{kJ mol\(^{-1}\)}\)

14.3(b)
\(\Delsub{f}H\st(\tx{AgCl, s})=-127.05\units{kJ mol\(^{-1}\)}\)

14.3(c)
\(S\m\st(\tx{AgCl, s})=96.16\units{J K\(^{-1}\) mol\(^{-1}\)}\)
\(\Delsub{f}S\st(\tx{AgCl, s})=-57.93\)\units{J K\(^{-1}\) mol\(^{-1}\)}
\(\Delsub{f}G\st(\tx{AgCl, s})=-109.78\units{kJ mol\(^{-1}\)}\)

14.4(b)
\(\Delsub{f}H\st(\tx{AgCl, s})=-126.81\units{kJ mol\(^{-1}\)}\)
\(\Delsub{f}G\st(\tx{AgCl, s})=-109.59\units{kJ mol\(^{-1}\)}\)

14.5
\(K\subs{s} = 1.76\timesten{-10}\)

14.6(b)
\(\g_{\pm}=0.756\)

14.7(b)
\(\Delsub{f}G\st=-210.72\units{kJ mol\(^{-1}\)}\)

14.7(c)
\(K\subs{s}=1.4\timesten{-18}\)

14.8
\(E\st = 0.071\V\)

14.9(c)
\(\Eeq\st=1.36\V\)

14.9(d)
In the cell:
\(\dq/\dif\xi = 2.27\units{kJ mol\(^{-1}\)}\)
In a reaction vessel:
\(\dq/\dif\xi = -259.67\units{kJ mol\(^{-1}\)}\)

14.9(e)
\(\dif\Eeq\st/\dif T = 3.9\timesten{-5}\units{V K\(^{-1}\)}\)