Problem 2

Dissociating gas cycle. Dinitrogen tetroxide forms an equilibrium mixture with nitrogen dioxide:

\ce{N2O4(g) <=> 2 NO2(g)}

. 1.00 mol of

\ce{N2O4}

was placed in an empty vessel of fixed volume 24.44 dm

^3

. The equilibrium gas pressure at 298 K was 1.190 bar; when heated to 348 K the equilibrium pressure rose to 1.886 bar. (a) Calculate

\Delta G^{\circ}

of the reaction at 298 K, assuming ideal gases. (b) Calculate

\Delta H^{\circ}

and

\Delta S^{\circ}

, assuming they do not vary significantly with temperature. The reversible dissociation of

\ce{N2O4}

can be exploited in power cycles: in step

3 \to 4

the hot gas mixture expands reversibly and adiabatically through a turbine. (c) Give the equation for the work done by 1 mol of an inert gas (air) during the reversible adiabatic expansion

3 \to 4

, assuming constant

C_{V,m}

and a temperature drop from

T_3

T_4

Step 1 of 5: Degree of dissociation at 298 K

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p_{ini} = \dfrac{RT}{V} = \dfrac{0.083145 \times 298}{24.44} = 1.014\ \text{bar}; \quad \dfrac{p_{eq}}{p_{ini}} = 1 + x \Rightarrow x = 0.174

Analysis

If $x$ moles of $\ce{N2O4}$ dissociate, $n_{tot} = 1 - x + 2x = 1 + x$ mol, so the pressure rises by the factor $1 + x$ : $x = p_{eq}/p_{ini} - 1 = 1.190/1.014 - 1 = 0.174$ .