# 14.15: Problems


Problems

1. When we express the energy of a system as a function of entropy, volume, and composition, we have $$E=E\left(S,V,n_1,n_2,\dots ,\ n_{\omega }\right)$$. Since $$S$$ and $$V$$ are extensive variables, we have $$\lambda E=E\left(\lambda S,\lambda V,{\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)$$. Find $${\left({\partial \left(\lambda E\right)}/{\partial \lambda }\right)}_{SV}$$. From this result, show that $G=\sum^{\omega }_{j=1}{{\mu }_jn_j}$

2. When we express the energy of a system as a function of pressure, temperature, and composition, we have $$E=E\left(P,T,n_1,n_2,\dots ,\ n_{\omega }\right)$$. Because P and T are independent of $$\lambda$$, $$\lambda E=E\left(P,T,{\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)$$. Show that

$E=\sum^{\omega }_{j=1} \overline{E}_jn_j$

3. From $$E\mathrm{=E}\left(P,T,n_{\mathrm{1}},n_{\mathrm{2}}\mathrm{,\dots ,\ }n_{\omega }\right)$$ and the result in problem 2, show that

$\left[{\left(\frac{\partial H}{\partial T}\right)}_P + P{\left(\frac{\partial V}{\partial T}\right)}_P\right]dT + \left[P{\left(\frac{\partial V}{\partial P}\right)}_T + T{\left(\frac{\partial V}{\partial T}\right)}_P\right]dP = \sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j$

Note that at constant pressure and temperature,

$\sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j\mathrm{=0}$

4. If pressure and temperature are constant, $$E=E\left(n_1,n_2,\dots ,\ n_{\omega }\right)$$ and $$\lambda E=E\left({\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)$$. Show that $$\sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j\mathrm{=0}$$ follows from these relationships.

5. A solution contains $$n_1$$ moles of component 1, $$n_2$$ moles of component 2, $$n_3$$ moles of component 3, etc. Let $$n=n_1+n_2+n_3+...$$ The mole fraction of component $$j$$ is $$x_j={n_j}/{n}$$. Show that $\left(\frac{\partial x_j}{\partial n_j}\right)=\frac{n-n_j}{n^2}$ and, for $$j\neq k$$, $\ \left(\frac{\partial x_j}{\partial n_k}\right)=\frac{-n_j}{n^2}$ What are $\left(\frac{\partial x_1}{\partial n_1}\right)$ and $\left(\frac{\partial x_2}{\partial n_2}\right)$ if the solution has only two components?

6. For any extensive state function, $$Y\left(P,T,n_1,n_2,\dots ,\ n_{\omega }\right)$$, the arguments developed in this chapter lead, at constant $$P$$ and$$\ T$$, to the equations

$Y=n_1{\overline{Y}}_1+n_2{\overline{Y}}_2+\dots +n_{\omega }{\overline{Y}}_{\omega }$ and $0=n_1d{\overline{Y}}_1+n_2d{\overline{Y}}_2+\dots +n_{\omega }d{\overline{Y}}_{\omega }$

Where $${\overline{Y}}_j$$ is the partial molar quantity $${\left({\partial Y}/{\partial n_j}\right)}_{P,T,n_{m\neq j}}$$.

(a) Prove that $$0=x_1d{\overline{Y}}_1+x_2d{\overline{Y}}_2+\dots +x_{\omega }d{\overline{Y}}_{\omega }$$

(b) Prove that $0=n_1\left(\frac{\partial {\overline{Y}}_1}{\partial n_1}\right)+n_2\left(\frac{\partial {\overline{Y}}_2}{\partial n_2}\right)+\dots +n_{\omega }\left(\frac{\partial {\overline{Y}}_{\omega }}{\partial n_{\omega }}\right)$ (c) Prove that $0=x_1\left(\frac{\partial {\overline{Y}}_1}{\partial x_1}\right)+x_2\left(\frac{\partial {\overline{Y}}_2}{\partial x_2}\right)+\dots +x_{\omega }\left(\frac{\partial {\overline{Y}}_{\omega }}{\partial x_{\omega }}\right)$

7. The enthalpy of mixing is measured in a series of experiments in which solid solute, $$A$$, dissolves to form an aqueous solution. These enthalpy data are represented well by empirical equations $${\Delta }_{mix}H={\alpha }_1\underline{m}+{\alpha }_2{\underline{m}}^2$$, $${\alpha }_1={\beta }_{11}+{\beta }_{12}\left(T-273.15\right)$$ and

$${\alpha }_2={\beta }_{21}+{\beta }_{22}\left(T-273.15\right)$$ with ${\beta }_{11}=10.0\ \mathrm{kJ}\ {\mathrm{molal}}^{-1}$ ${\beta }_{12}=-0.14\ \mathrm{kJ}\ {\mathrm{molal}}^{-2}\ K^{-1}$ ${\beta }_{21}=-3.00\ \mathrm{kJ}\ {\mathrm{molal}}^{-1}$ ${\beta }_{22}=-0.040\ \mathrm{kJ}\ {\mathrm{molal}}^{-2}\ K^{-1}$ Find $${\overline{L}}_A$$, $${\overline{L}}_{H_2O}$$, $${\overline{J}}_A$$, and $${\overline{J}}_{H_2O}$$ as functions of $${\underline{m}}_A$$ and $$T$$. Find $${\overline{L}}_A$$, $${\overline{L}}_{H_2O}$$, $${\overline{J}}_A$$, and $${\overline{J}}_{H_2O}$$ for a one molal solution at 209 K. What is the value of

${ \ln \frac{{\tilde{a}}_A\left(1\mathrm{\ molal},290\mathrm{\ K}\right)}{{\tilde{a}}_A\left(1\mathrm{\ molal},273.15\mathrm{\ K}\right)}\ }$

Notes

$${}^{1}$$ We can make other assumptions. It is possible to describe an inhomogeneous system as a collection of many macroscopic, approximately homogeneous regions.

14.15: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.