[TOP/TH-PSOLV] Problem Solving in Thermodynamics

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 \(P V = nRT\) and for van der
Wall gas is
\[\begin{equation*} \Big(p + \frac{an^2}{V^2}\Big )(V − nb) = nRT \end{equation*}\]

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 \(T + ∆T\) ? The coefficients of linear expansion of the two metals are \(α_1\) and \(α_2\) ,
respectively, with \(α_1 > α_2\). Assume each metal has thickness \(x/2\), and that \(x << R.\)

Let
\[\frac{\partial (x,y)}{\partial (a,b)}\,\equiv\,\left|\begin{array}{ll} \frac{\partial x}{\partial a}&\frac{\partial y}{\partial a}\\ \frac{\partial x}{\partial b}&\frac{\partial y}{\partial b}\\ \end{array}\right|\]

Then show that
\[\frac{\partial (x,y)}{\partial (a,b)}\frac{\partial (a,b)}{\partial (c,d)}\,=\,\frac{\partial (x,y)}{\partial (c,d)}\]

Remarks : 1. This can be generalised to higher dimensions.

2. This can be found in books - and is very useful in changing variables in multiple integrals.

Express the temperatures given below in \(^\circ C,R\), and \(^\circ F\). Write your answers in form of table.

• (a) Triple point of Hydrogen 13.81 K
• (b) Boiling point of Ne 27.102 K
• (c) Boiling point of Oxygen 90.188 K
• (d) Melting point of zinc 692.664
• (e) Melting point of gold 1337.58

A new scale is to defined in terms of ideal gas scale of
temperature. Let the difference of ice points and the steam points
be fixed at \(t_s-t_i=180\) deg. If
\(\displaystyle{\lim_{P_{TP}\to0}\left({P_s\over P_i}\right)}\) is
given to be 1.3661 as the best experimental value, find \(t_s\) and
\(t_i\) on the new scale.

In the table below, a number in the top row represents the
pressure of a gas in the bulb of a constant volume gas thermometer
when the bulb is immersed in the triplet cell. The bottom row
represents the corresponding readings of the pressure when the bulb
is surrounded by a material at constant unknown temperature.
Calculate the ideal gas temperature of this material (use five
significant figures) {Zemansky}

 Is it necessary to convert the

pressures from mm Hg to Pascal?