Chapter 10, Production of Low Temperatures and their Applications Video Solutions, Thermal Physics : Kinetic Theory, Thermodynamics and Statistical Mechanics

Problem 1

In the region of moderate pressures, the equation of state of one mole of a gas may be written as
$$
p V=R T+c_{2} p+c_{3} p^{2}
$$
where $c_{2}$ and $c_{3}$ are the second and third virial coefficients. Taking these to be functions of $T$ only, show that as the pressure approaches zero
$$
\mu c_{p} \rightarrow T \frac{\mathrm{d} c_{2}}{\mathrm{~d} T}-c_{2}
$$
Also show that the inversion curve is described by equation
$$
p\left(c_{3}-T \frac{\mathrm{d} c_{3}}{\mathrm{~d} T}\right)+\left(c_{2}-\frac{\mathrm{d} c_{2}}{\mathrm{~d} T}\right)=0
$$

Narayan Hari

Numerade Educator

01:30

Problem 2

We know that $\mu$ provides us with a measure of temperature change during a throttling process. A similar measure of the temperature change produced by an isentropic change of pressure can be obtained from the coefficient $\mu_{S}$ defined as
$$
\mu_{S}=\left(\frac{\partial T}{\partial p}\right)_{s}
$$
Prove that
$$
\mu_{S}-\mu=\frac{V}{C_{p}}
$$

Sheh Lit Chang

University of Washington

View

Problem 3

Calculate the pressure corresponding to an inversion temperature of $20 \mathrm{~K}$ for helium assuming that it behaves as a van der Waals' gas. Use the data given in Exercise 3.11.

Mirza Aslam Beig

Numerade Educator

05:28

Problem 4

Using the fact that $\left(\frac{\partial S}{\partial V}\right)_{T=0}=\left(\frac{\partial^{2} S}{\partial V^{2}}\right)_{T=0}=0$ at $T=0$,
prove that $\lim _{T \rightarrow 0}\left(\frac{\partial\left(1 / \beta_{T}\right)}{\partial T}\right)_{V}=0$ where $\beta_{T}$ is isothermal compressibility.

Gaurav Gupta

Numerade Educator

01:50

Problem 5

Calculate the fall in temperature produced by adiabatic demagnetisation of a paramagnetic salt, if magnetic field is reduced from 8400 Oe to zero at $3 \mathrm{~K}$. Take Curie constant per $\mathrm{cm}^{3}=0.6 \mathrm{erg} \mathrm{K} \mathrm{g}^{-1} \mathrm{Oe}^{-2}$ and $C_{B}=0.2 \mathrm{cal} \mathrm{g}^{-1} \mathrm{~K}^{-1}$.

Anand Jangid

Numerade Educator