# 8.10: Problems

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1. One mole of an ideal gas reversibly traverses Cycle I above. Step a is isothermal. Step b is isochoric (constant volume). Step c is isobaric (constant pressure). Assume $$C_V$$ and $$C_P$$ are constant. Find $$q$$, $$w$$, $$\Delta E$$, and $$\Delta H$$ for each step and for the cycle. Prove $$C_P=C_V+R$$.

2. One mole of an ideal gas reversibly traverses Cycle II below. Step a is the same isothermal process as in problem 1. Step d is adiabatic. Step e is isobaric. Assume $$C_V$$ and $$C_P$$ are constant. Find $$q$$, $$w$$, $$\Delta E$$, and $$\Delta H$$ for each step and for the cycle.

3. One mole of an ideal gas reversibly traverses Cycle III below. Step a is the same isothermal process as in

problem 1. Step f is adiabatic. Step g is isochoric. Assume $$C_V$$ and $$C_P$$ are constant. Find $$q$$, $$w$$, $$\Delta E$$, and $$\Delta H$$ for each step and for the cycle.

4. One mole of an ideal gas reversibly traverses Cycle IV. Step h is isobaric. Step f is the same adiabatic process as in problem 3. Step i is isochoric. Assume $$C_V$$ and $$C_P$$ are constant. Find $$q$$, $$w$$, $$\Delta E$$, and $$\Delta H$$ for each step and for the cycle.

5. Prove that the work done on the system is positive when the system traverses Cycle I. Note that Cycle I traverses the region of the $$PV$$ plane that it encloses in a counter-clockwise direction. Hint: Note that $$T_2<t_1$$. Show that $${V_2}/{V_1}={T_2}/{T_1}$$.

6. Cycles III and IV share a common adiabatic step. Express the work done in each of these cycles in terms of $$V_1$$, $$V_2$$, and $$T_1$$. Prove that the work done in Cycle IV is greater than the work done in Cycle III.

7. Cycles I, II, and III share a common first step, a. Express $$V_3$$, $$T_3$$, and $$T_4$$ in terms of $$V_1$$, $$V_2$$, and $$T_1$$. For $$V_1=10\ \mathrm{L}$$, $$V_2=2\ \mathrm{L}$$, and $$T_1=400\ \mathrm{K}$$, show that the work done decreases in the order Cycle I $$\mathrm{>}$$ Cycle III $$\mathrm{>}$$ Cycle II.

8. For water, the enthalpies of fusion and vaporization are $$6.009$$ and $$40.657\ \mathrm{k}\mathrm{J}\ {\mathrm{mol}}^{-1}$$, respectively. The heat capacity of liquid water varies only weakly with temperature and can be taken as $$\mathrm{75.49\ }\mathrm{J}\ {\mathrm{mol}}^{-1}\ {\mathrm{K}}^{-1}$$. The heat capacity of water vapor varies with temperature: $C_P\left(H_2O\mathrm{,\ g}\right)=30.51+\left(1.03\times {10}^{-2}\right)T \nonumber$ where $$T$$ is in degrees K and the heat capacity is in $$\mathrm{J}\ {\mathrm{mol}}^{-1}\ {\mathrm{K}}^{-1}$$. Estimate the enthalpy of sublimation of water.

9. If we truncate the virial equation $$\left(Z=1+B^*\left(T\right)P+\dots \right)$$ and make use of $$B\left(T\right)=RTB^*\left(T\right)$$, where$$\ B\left(T\right)$$ is the “second virial coefficient” most often given in data tables, the molar volume is $\overline{V}=\frac{RT}{P}+B\left(T\right) \nonumber$ Show that ${\left(\frac{\partial H}{\partial P}\right)}_T=B\left(T\right)-T\left(\frac{dB}{dT}\right) \nonumber$

The Handbook of Chemistry and Physics (CRC Press, 79$${}^{th}$$ Ed., 1999, p. 6–25) gives the temperature dependence of $$B$$ for water vapor as

$B=-1158-5157t-10301t^2-10597t^3-4415t^4 \nonumber$

where $$t=\left({298.15}/{T}\right)-1$$, $$T$$ is in degrees kelvin, and the units of $$B$$ are $${\mathrm{cm}}^{-3\ }{\mathrm{mol}}^{-1}$$. Estimate the enthalpy change when one mole of water vapor at 1 atm and 100 C is expanded to the equilibrium sublimation pressure, which for this purpose we can approximate as the triple-point pressure, $$610\ \mathrm{Pa}$$. How does this value compare to the result of problem 8?

10. The heat capacities of methanol liquid and gas are $$81.1$$ and $$44.1\ \mathrm{J}\ {\mathrm{mol}}^{-1}\ {\mathrm{K}}^{-1}$$, respectively. The second virial coefficient for methanol vapor is

$B=-1752-4694t \nonumber$

where $$t=\left({298.15}/{T}\right)-1$$, $$T$$ is in degrees kelvin, and the units of $$B$$ are $${\mathrm{cm}}^{-3\ }{\mathrm{mol}}^{-1}$$. Referring to the discussion of methanol vaporization in §5, calculate $${\Delta }_{\left(1\right)}H$$, $${\Delta }_{\left(4\right)}H$$, $${\Delta }_{\left(5\right)}H$$, $${\Delta }_{\left(vap\right)}H^o$$. Compare this value of $${\Delta }_{\left(vap\right)}H^o$$ to the value given in the text. [Data from the Handbook of Chemistry and Physics, CRC Press, 79$${}^{th}$$ Ed., 1999, p. 5-27 and p. 6-31.]

 Molecular formula Name $${\Delta }_fH^o$$$$\left(\mathrm{k}\mathrm{J}\ {\mathrm{mol}}^{-1}\right)$$ $$H_2O\ \left(\mathrm{liq}\right)$$ Water $$-285.8$$ $$CO\ \left(\mathrm{g}\right)$$ Carbon monoxide $$-110.5$$ $$CO_2\ \left(\mathrm{g}\right)$$ Carbon dioxide $$-393.5$$ $$CH_4\left(\mathrm{g}\right)$$ Methane $$-74.6$$ $$C_2H_4\left(\mathrm{g}\right)$$ Ethylene $$52.4$$ $$C_2H_6\left(\mathrm{g}\right)$$ Ethane $$-84.0$$ $$CH_3CH_2OH\ \left(\mathrm{liq}\right)$$ Ethanol $$-277.6$$ $$CH_3CHO\ \left(\mathrm{liq}\right)$$ Acetaldehyde $$-192.2$$ $$CH_3CO_2H\ \left(\mathrm{liq}\right)$$ Acetic acid $$-484.3$$ $$CH_3CH_2CHO\ \left(\mathrm{liq}\right)$$ Propanal $$-215.6$$ $$C_6H_6\ \left(\mathrm{liq}\right)$$ Benzene $$49.1$$ $$C_6H_5CO_2H\ \left(\mathrm{s}\right)$$ Benzoic acid $$-385.2$$

11. Using data from the table above, find the enthalpy change for each of the following reactions at 298 K.

(a) $$C_2H_6\left(\mathrm{g}\right)+ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/08:_Enthalpy_and_Thermochemical_Cycles/8.10:_Problems), /content/body/p[19]/span, line 1, column 1  \ O_2\left(\mathrm{g}\right)\to CH_3CH_2OH\left(\mathrm{liq}\right)$$

(b) $$C_2H_4\left(\mathrm{g}\right)+ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/08:_Enthalpy_and_Thermochemical_Cycles/8.10:_Problems), /content/body/p[20]/span, line 1, column 1  \ O_2\left(\mathrm{g}\right)\to CH_3CHO\left(\mathrm{liq}\right)$$

(c) $$C_2H_6\left(\mathrm{g}\right)+ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/08:_Enthalpy_and_Thermochemical_Cycles/8.10:_Problems), /content/body/p[21]/span, line 1, column 1  \ O_2\left(\mathrm{g}\right)\to CH_3CHO\left(\mathrm{liq}\right)+H_2O\left(\mathrm{liq}\right)$$

(d) $$C_6H_6\left(\mathrm{liq}\right)+\ CO_2\left(\mathrm{g}\right)\to C_6H_5CO_2H\left(\mathrm{s}\right)$$

(e) $$CH_3CHO\left(\mathrm{liq}\right)+ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/08:_Enthalpy_and_Thermochemical_Cycles/8.10:_Problems), /content/body/p[23]/span, line 1, column 1  \ O_2\left(\mathrm{g}\right)\to CH_3CO_2H\left(\mathrm{liq}\right)$$

(f) $$CH_4\left(\mathrm{g}\right)+H_2O\left(\mathrm{liq}\right)\to CO\left(\mathrm{g}\right)+3\ H_2\left(\mathrm{g}\right)$$

(g) $$CH_4\left(\mathrm{g}\right)+H_2O\left(\mathrm{liq}\right)+ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/08:_Enthalpy_and_Thermochemical_Cycles/8.10:_Problems), /content/body/p[25]/span, line 1, column 1  \ O_2\left(\mathrm{g}\right)\to CO_2\left(\mathrm{g}\right)+3\ H_2\left(\mathrm{g}\right)$$

(h) $$C_2H_4\left(\mathrm{g}\right)+CO\left(\mathrm{g}\right)+\ H_2\left(\mathrm{g}\right)\to CH_3CH_2CHO\left(\mathrm{liq}\right)$$

Notes

$$^{1}$$ Data compiled by The Committee on Data for Science and Technology (CODATA) and reprinted in D. R. Linde, Editor, The Handbook of Chemistry and Physics, 79$${}^{th}$$ Edition (1998-1999), CRC Press, Section 5.

$${}^{2}$$ D. R. Linde, op. cit., p. 6-104.

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