Question 3.4: In Joule’s experiment to determine (∂U/∂U)T, , the heat capa...
In Joule’s experiment to determine \left(\partial U/\partial V \right)_{T} , the heat capacities of the gas and the water bath surroundings were related by C_{surroundings}/C_{system}\approx 1000. If the precision with which the temperature of the surroundings could be measured is \pm 0.006°C, what is the minimum detectable change in the temperature of the gas?
View the experimental apparatus as two interacting systems in a rigid adiabatic enclosure. The first is the volume within vessels A and B, and the second is the water bath and the vessels. Because the two interacting systems are isolated from the rest of the universe,
q =C_{water bath}\Delta T_{water bath}+C_{gas}\Delta T_{gas}=0\Delta T_{gas}=-\frac{C_{water bath}}{C_{gas}}\Delta T_{water bath}=- 1000×\left( \pm 0.006 °C \right)=\pm 6°C
In this calculation, \Delta T_{gas} is the temperature change that the expanded gas undergoes to reach thermal equilibrium with the water bath, which is the negative of the temperature change during the expansion. Because the minimum detectable value of \Delta T_{gas} is rather large, this apparatus is clearly not suited for measuring small changes in the temperature of the gas in anexpansion.