Kevin S. Huang

2011: Problem 21

Topic: Other
Concepts: Dimensional analysis

Solution:

We can find the form of $n(V_m, T, R, \sigma)$ via dimensional analysis. We have the units of all variables:

[n]=\mathrm{mol}

[V_m]=\mathrm{m^3}

[T]=\mathrm{K}

[R]=\mathrm{\frac{J}{K \cdot mol}}

[\sigma]=\mathrm{\frac{N}{m^2}}

Since $[n]=\mathrm{mol}$ , we must have $n \propto R^{-1}$ . Then

[R^{-1}]=\mathrm{\frac{mol \cdot K}{J}}

so we have to divide by $T$ to cancel out $\mathrm{K}$ and obtain $n \propto R^{-1}T^{-1}$ . Then

[R^{-1}T^{-1}]=\mathrm{\frac{mol}{J}}

so we have to get rid of units of $\mathrm{J}$ . If we multiply $\sigma$ and $V_m$ , we find

[\sigma V_m]=\mathrm{N \cdot m}=\mathrm{J}

so we can just multiply $\sigma V_m$ to the earlier expression. Then

[\sigma V_mR^{-1}T^{-1}]=\mathrm{mol}

as required. Therefore,

n \propto \frac{\sigma V_m}{RT}

so the answer is A.