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Thermal Physics
Problem sheet 7
Each question carries 5 marks
2nd November 2021 Due 9th November 2021
19. Consider a 2 dimensional phase space ( $q,p$) with a rectangular region defined by four corners as shown.
If the region ABCD is the phase space region at time time t = 0 , find the region $A'B'C'D'$ at time t given the Hamiltonian is
$$ H\,=\,\frac{p^2}{2m}\,-\, m a q $$
and explicitly verify that the area is constant. Take the coordinates of A,B,C and D as $(q_A,p_A)\,,\,(q_B,p_A)\,,\,(q_B,p_C)$ and $(q_A,p_C)$ respectively
20. Using the identity $\Gamma(N\,+\,1)\,=\,N!=\,\int_0^\infty x^Ne^{-x} dx $
and write the integral in the form
$$ \int_0^\infty e^{Ng(x)}dx $$
Find the maximum of $g(x)$ ( which occurs say at $x_0$ ). Assume the integral
is dominated by the contribution from the neighbourhood of $x_0$ for large N. Expanding $g(x)$ up to second order in $(x-x_0)$ derive the Stirling's approximation ( for $n\rightarrow\,\infty$)
$$ \rm{ln}N!\,=\,N\rm{ln}N\,-\,N\,+\,\frac{1}{2}\rm{ln}(2\pi N)$$
( More exact formula , just for information, is
$$ N!\,=\,\left(\sqrt{2\pi}\right)e^{-N}N^{N+1/2}\left[1\,+\,\frac{1}{12N}\,+\,\frac{1}{288N^2}\,+\,\ldots\right]$$
)
21.Show that the internal energy of a material whose equation of state is of the form
$$ P\,=\,f(V)T $$
is independent of the volume V. ( P,T are pressure and temperature)
16. Consider a paramagnetic system, with variables magnetization $M$, the magnetic field $B$ and absolute temperature $T$. ( We assume it's dependence on pressure as negligible). The equation of state is ( which will be obtained from statistical mechanics later in the course) is
$$ M\,=\,C\frac{B}{T}, $$
where $C$ is a constant ( referred to as the Curie constant, who had experimentally obtained this relation.
The system's internal energy is ( for a one-dimensional system)
$$ U\,=\,-MB.$$
The work done on the system by external surrounding is $-MdB$
(a) Write the expression for $DQ$ in terms of $dM$ and $dB$
(b) Write the equation for entropy change $dS$ in terms of $dM$ and $dB$
(c) Obtain the entropy $S$
17. Consider a box of volume $V$ containing $N$ non-interacting free particles.
Consider a small volume $v$ inside the box. What is the probability that there are $n$ particles inside the volume $v$ ans $N\,-\,v$ in the rest of the box ( having volume $V\,-\,v$ ?. Maximize the probability and show that the maximum occurs for $ \bar{n}\,=\,\frac{N}{V}\times v$. ( This proves that the probability for the density to be uniform is the highest)
18. Consider a one-dimensional harmonic oscillator of mass $m$ and whose total energy is given by
$$ E\,=\,\frac{p^2}{2m}\,+\,\frac{m\omega^2x^2}{2} $$
( The notation is the standard one). Find the volume in phase space for the system to be in the energy range $U$ and $U\,+\,\Delta U$
Reading Assignment 1 : First six sections of the class XII NCERT book, Chapter 12 ( Thermodynamics)
Reading assignment 2: From the book Principles of Thermodynamics by N.D.Hari Dass CRC Press Section 1.1 ( discusses thermometry first six pages)
1. Show that constant volume thermometers using an ideal gas as
well as a van der Waals gas both yield the same temperature scale when uniform
scales are adopted.
Equation of state for ideal gas is $PV\,=\,nRT$ and for van der Wall gas is
$$ \left(p\,+\,\frac{an^2}{V^2}\right)\left(V\,-\,nb\right)\,=\,nRT $$
2. A bimetallic strip of total thickness x is straight at temperature T.
What is the radius of curvature of the strip, $R$, when it is heated to
temperature $T\,+\,\Delta T$? The coefficients of linear expansion of the two metals are $\alpha_1$ and $\alpha_2$, respectively, with $\alpha_1\, >\,\alpha_2$. Assume each metal has thickness $x/2$, and that $x <<\, R$.
3. Which of the following quantities are extensive and which are intensive?
(a) The magnetic moment of a gas.
(b) The electric field E in a solid.
(c) The length of a wire.
(d) The surface tension of an oil film.
