Ideal Gas Thermometer

Introduction

If we heat an ideal gas it will want to expand or increase its pressure. If we want to have a measure of its new temperature we could either employ a pressure gauge or we could have the expansion raise a movable piston. The simplest implementation of the gas thermometer is movable piston with the working gas trapped below a movable piston which has a vacuum space above it. The temperature of the gas is changed by adding heat to it. This results in an increase in internal kinetic energy. But when the piston raises to expand the gas back to its old pressure, the gas gives up some of this heat because the atoms that hit the rising piston have their vertical speed reduced. The following calculations will document how much the energy is reduced during expansion.

Calculation of Final Energy and Pressure

To start we will assume a 2 dimensional gas which has a simple ideal gas equation.

Before adding internal energy δU we have the ideal gas law:

$P_{0} V_{0} = U_{0}$

(1.1)

When we add internal energy δU to the gas this expression changes to:

$P_{1} V_{0} = U_{0} + δ U \equiv U_{1}$

(1.2)

After expanding the volume to V₂so that the pressure returns to P₀we have the following equation :

$P_{0} V_{2} = U_{2}$

(1.3)

where

$U_{2} = U_{0} + δ U - \int_{V_{0}}^{V_{2}} P (V) d V$

(1.4)

Problem is that we don't know P(v) in the integral.

Let's assume that

$U = U_{1} \frac{V_{0}}{V}$

(1.5)

where U₁ =U₀+δU.

Then using the expression for U₂:

$P_{0} V_{2} = (U_{0} + δ U) \frac{V_{0}}{V_{2}}$

(1.6)

Then we may solve for V₂ and obtain:

$V_{2} = \sqrt{\frac{U_{1} V_{0}}{P_{0}}} = \sqrt{\frac{(P_{0} V_{0} + δ U) V_{0}}{P_{0}}}$

(1.7)

Then U₂ is:

$U_{2} = \frac{U_{1} V_{0}}{V_{2}} = \frac{U_{1} V_{0}}{\sqrt{\frac{U_{1} V_{0}}{P_{0}}}} = \sqrt{U_{1} P_{0} V_{0}} = \sqrt{U_{1} U_{0}}$

(1.8)

The general expression for P during the expansion is:

$P = \frac{U}{V} = \frac{U_{1} V_{0}}{V^{2}}$

(1.9)

Conclusion

It is important to note that the expressions for V₂, U₂ and general expressions for U and P above agree with the plotted results for the ballistic gas animation.