Riemann surface
matlab

Consider the following equations for the pressure of a real gas. For each equation, find the dimensions of the constants $a$ and $b$ and express these dimensions in SI units.
(a) The Dieterici equation:
$$p=\frac{R T e^{-(a n / V R T)}}{(V / n)-b}$$

Solution:
Since $a n / V R T$ is a power, it is dimensionless and $a$ has the same dimensions as $V R T / n$. These dimensions are volume – energy/amount ${ }^2$, expressed in $\mathrm{m}^3 \mathrm{~J} \mathrm{~mol}^{-2} . b$ has the same dimensions as $V / n$, which are volume/amount expressed in $\mathrm{m}^3 \mathrm{~mol}^{-1}$.

(b) The Redlich-Kwong equation:
$$p=\frac{R T}{(V / n)-b}-\frac{a n^2}{T^{1 / 2} V(V+n b)}$$

Solution:
The term $a n^2 / T^{1 / 2} V(V+n b)$ has the same dimensions as $p$, so $a$ has the same dimensions as $T^{1 / 2} V^2 p n^{-2}$. The SI units are $\mathrm{K}^{1 / 2} \mathrm{~m}^6 \mathrm{~Pa} \mathrm{~mol}^{-2} \cdot b$ has the same dimensions as $V / n$, which are volume/amount expressed in $\mathrm{m}^3 \mathrm{~mol}^{-1}$.

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