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Consider a cylindrical system of ideal gas with a movable piston as shown . All the wall.\, including the piston are thermally insulating. A weight W is placed on the piston. Treating this system as a closed one, Show that $$ \frac{W}{A}\,=\,\frac{2E}{3V}$$ where A is the area of cross section of the cylinder, E the energy and V the volume of gas.

HSMani

A box of volume, \(V = L^3\) , contains an ideal gas of \(N\) identical atoms, each of which has spin, \(s = 1/2\), and magnetic moment, \(\mu\). A magnetic field, \(B\) is applied to the system. (a) Compute the partition function for this system. (b) Compute the internal energy and the heat capacity. (c) What is the magnetization?

Show that in the canonical ensemble formalism, the entropy $S$ of the system is related to the partition function $Q$ as given below. $$S=k_B\left[\ln Q+T\left(\frac{\partial\ln Q}{\partial T}\right)_V\right].$$}

KPN

The canonical partition function of a system of $N$ hypothetical particles each of mass $m$, confined to a volume $V$ at temperature $T$ is given by, $$Q(T,V,N) = V^N\left(\frac{2\pi k_B T}{m}\right)^{5N/2}.$$ Determine the equation of state of the hypothetical system. Also find $C_V$ - heat capacity at constant volume. Identify the hypothetical system. How many degrees of freedom does each particle of the hypothetical system have ?

KPN

\(\newcommand{\Prime}{^\prime}\)Drive \(3\times 3\) rotation matrix corresponding to a rotation about axis \(\hat{n}\) by an angle \(\theta\) using the formula

\begin{equation}

\vec{x}\Prime = (\vec{x}\cdot\hat{n})\hat{n} -

(\hat{n}\times\vec{x})\sin\theta -

\hat{n}\times(\hat{n}\times\vec{x})\cos\theta.

\end{equation}

Using definition of Levi-Civita epsilon symbol and Kronecker delta symbol to show that \begin{equation}\epsilon_{i\,j\,k}\,\epsilon_{k\,\ell\,m}\ =\ (\,\delta_{i\ell}\,\delta_{jm}\,-\,\delta_{im}\,\delta_{j\ell}) \end{equation} Use this identity to prove that \begin{equation}\vec{A}\times(\vec{B}\times\vec{C})=(\vec{A}\cdot\vec{C})\vec{B} -(\vec { A } \cdot\vec{B})\vec{C}.\end{equation}

- Consider a system of three point masses connected by rigid rods. Find the number of constraints and hence the number of generalized coordinates required for this system when the masses do not lie in a straight line.

- Next repeat the above problem for three three masses constrained to lie in a straight line.

- Count the number of constraints and hence find the number of degrees of a freedom of a rigid body consisting of \(n\) point masses separated by fixed distances. Compute the value of number of degrees of freedom for a few values of \(n\) . What happens when \(n\) is very large?

A useful reference for counting of degrees of freedom for a rigid body, in a manner described above, is

Jorge Bernal et al., "*Exact calculation of the number of degrees of freedom of a rigid body* *constituted by n particles*". arXiv:1002.2002v1 (Feb9,2010)

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