Question

Consider a classical system of $N$ point particles of mass $m$ in a volume $V$ at temperature $T$. Let $U$ be the total energy of the system, $p$ the pressure. The particles interact through a two-body central potential $$ \phi\left(r_{i j}\right)=\frac{A}{r_{i j}^n}, \quad A>0, \quad n>0, \quad r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right| . $$ Notice the scaling property $\phi(\gamma r)=\gamma^{-n} \phi(r)$ for any $\gamma$. From this, and from scaling arguments (e.g. applied to the partition function) show that $$ U=a p V+b N k T, \quad k=\text { Boltzmann's const. }, $$ where the constants $a$ and $b$ depend on the exponent $n$ in the pairwise potential. Express $a$ and $b$ in terms of $n$.

   Consider a classical system of $N$ point particles of mass $m$ in a volume $V$ at temperature $T$. Let $U$ be the total energy of the system, $p$ the pressure. The particles interact through a two-body central potential
$$
\phi\left(r_{i j}\right)=\frac{A}{r_{i j}^n}, \quad A>0, \quad n>0, \quad r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right| .
$$
Notice the scaling property $\phi(\gamma r)=\gamma^{-n} \phi(r)$ for any $\gamma$. From this, and from scaling arguments (e.g. applied to the partition function) show that
$$
U=a p V+b N k T, \quad k=\text { Boltzmann's const. },
$$
where the constants $a$ and $b$ depend on the exponent $n$ in the pairwise potential. Express $a$ and $b$ in terms of $n$.

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 146

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Step 1

The particles interact through a two-body central potential given by \[ \phi(r_{ij}) = \frac{A}{r_{ij}^n} \] where $ A > 0 $ and $ n > 0 $. The distance between particles $ i $ and $ j $ is $ r_{ij} = |\mathbf{r}_i - \mathbf{r}_j| $. Show more…

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Consider a classical system of $N$ point particles of mass $m$ in a volume $V$ at temperature $T$. Let $U$ be the total energy of the system, $p$ the pressure. The particles interact through a two-body central potential $$ \phi\left(r_{i j}\right)=\frac{A}{r_{i j}^n}, \quad A>0, \quad n>0, \quad r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right| . $$ Notice the scaling property $\phi(\gamma r)=\gamma^{-n} \phi(r)$ for any $\gamma$. From this, and from scaling arguments (e.g. applied to the partition function) show that $$ U=a p V+b N k T, \quad k=\text { Boltzmann's const. }, $$ where the constants $a$ and $b$ depend on the exponent $n$ in the pairwise potential. Express $a$ and $b$ in terms of $n$.

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