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Consider a system of $N$ atoms. Assuming that they can exist in two states only.
The ground state having energy zero and an excited state having energy \(\epsilon\).

1. Find the number of micro states with total energy \(U\).
2. Write an expression for entropy and using Stirling approximation for the factorial \[ \ln (N!) \approx N \ln N - N\] find the temperature of the system and hence show that \[U = \frac{N\epsilon}{1+ e^{\epsilon/kT}}\] What is fraction  of atoms are in the excited state at very large temperature \((T >> kT)\)?

A thermally insulated chamber contains $1000$ moles of mono-atomic ideal gas\footnote{$PV=nRT$} at $10$ atm. pressure. Its temperature is $300$ K. The gas leaks out slowly through a valve into the atmosphere. The leaking process is quasi static, reversible and adiabatic \(^\)1

1. How many moles of gas shall be left in the chamber eventually?
2. What shall be the temperature of the gas left in the chamber ?

$1$ atm $=0.981\times 10^6$ Pa ; $\gamma={\displaystyle \frac{C_P}{C_V}=\frac{5}{3}}$ .       

Footnote 1 :: $PV^\gamma=\Theta$ where $\Theta$ is a constant.}.                                        KPN

A particular system obeys the fundamental equation, $$U=A\ \frac{N^3}{V^2}\exp\left( \frac{S}{Nk_B}\right), $$ where $A$ (joule metre$^2$) is a constant. Initially the system is at $T=317.48$ kelvin, and $P=2\times 10^5$ pascals. The system expands reversibly until the pressure drops to $10^5$ pascals, by a process in which the entropy does not change. What is the final temperature ?                                                                                 

                                    KPN

Consider one mole of an ideal gas. A molecule of the gas is mono atomic and is spherically symmetric. The system of ideal gas is in equilibrium. Its temperature $T_1=300$ k and its volume $V_1=1$ litre. Since the system is in equilibrium, it can be represented by a point $A=( V_1,T_1)$, in the temperature-volume thermodynamic phase plane. We are taking Volume along the $X$ axis and Temperature along the $Y$ axis.

Now consider a process by which the system expands to a volume $V_2=2$ litres. The process is adiabatic.

• {\bf Case-1 :} The process is quasi static and reversible. Let $T_2$ be the temperature of the system at the end of the process. Let $B=(V_2,T_2)$ denote the system at the end of the process. {\bf Find} $T_2$. The process $A\to B$ can be represented by a curve joining $A$ to $B$. The curve is called an adiabat.
• {\bf Case-2 :} The process is not reversible. Hence the process can not be represented by a curve in the thermodynamic phase diagram. The system disappears from $A$ at the start of the process. At the completion of the process, if we wait long enough, the system would equilibrate and appear at a point $B^\prime=(T_2^\prime,V_2)$ in the phase diagram. There is no ready-made formula for calculating $T_2^\prime$. Nor is there a formula for calculating the increase in entropy of the system in the irreversible process. Also these quantities depend on how far away the irreversible process is from its reversible companion. However $B^\prime$ is on a line parallel to $Y$ axis and passing through $B$. Employing the Second law of thermodynamics

Answer the following questions.

1. Find if the point $B^\prime$ is vertically above or below the point $B$.
2. Calculate the increase in entropy in terms of $T_2^\prime$.

The article ( Physics Today 32,(2000) by Elliot Lieb and Jacob Yngvason  titled " A Fresh Look at Entropy and the Second Law of Thermodynamics") write "... We use it to calculate entropy, specific and latent heat, phase transition properties, transport coefficients , and so on,with great accuracy.

Important examples abound, such as Max Planck's realization that by staring into the furnace he could find Avogadro's number and Linus Pauling's highly accurate back of the envelope calculation of the residual entropy of Ice.

" What method could Max Planck used to get the Avogadro number?

(  Linus Pauling's calculation will be discussed later.)

Please read the article by Lieb and Yngvason, it only 6 page long.

HSMani

Show from calculating
$$ \frac{\Delta S}{\Delta E}\,=\,\frac{1}{T}\,=\,k\beta$$
for the case of Fermi-Dirac distribution,
$$ \Omega_{FD}\,=\,\Pi_{i=1}^\infty\left({}^{g_i}C_{N_i}\right) $$
[ notation as used in the class ]. Also for the case of equilibrium we have
$$ N_i\,=\,\frac{g_i}{e^{\alpha+\beta\epsilon_i}\,+\,1} $$
Hint: Consider changes only in two levels say with energies $\epsilon _1$ and $\epsilon_2$. Then argue that the result so obtained is independent of the choice of levels]

NOTES FOR LECTURES ON CLASSICAL MECHANICS