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Thermal Physics.
Problem sheet 6.
Each question carries 5 marks
26th October 2021 Due 2nd November 2021
16. Find the specific heat $C_H$ of a substance at constant enthalpy and show that it is equal to $$ C_H\,=\,\frac{C_p}{1\,-\,\frac{T}{V}\times\left(\frac{\partial V}{\partial T}\right)_p} $$ where we are using the standard notation.
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.
