Irreversible Processes with Application to Helium II and the Knudsen Effect in Gases

Abstract

It is generally conceded that the pressure difference $\mathrm{dp}$ which develops in a steady state, when two portions of helium II are kept at different temperatures and separated by a capillary, is given by $\frac{\mathrm{dp}}{\mathrm{dT}}=\frac{Q}{\mathrm{TV}}$, where $Q$ is the heat absorbed as volume $V$ leaves the capillary at absolute temperature $T$. London gives $Q=TS$ where $S$ is the entropy, for the case where no heat is absorbed along the capillary ($c=0$), while Gorter gives $Q=T{x}_{1}(\frac{\ensuremath{\partial}S}{\ensuremath{\partial}{x}_{1}})$ where ${x}_{1}$ is the mole fraction of normal helium in the helium II. Gorter's equation coincides with London's, which is reasonably well confirmed experimentally, only if ${\overline{S}}_{2}$ the partial molal entropy of superfluid is zero. In this paper an analysis of the situation, including a careful discussion of the meaning of $Q$, is attempted, and it is shown that for $c=0$ London's equation follows from the equation for $\frac{\mathrm{dp}}{\mathrm{dT}}$ and the conservation of energy. The equations which are developed are quite general, and can be applied directly to other cases, for example, the Knudsen effect in gases. They still do not permit an answer to the question concerning ${\overline{S}}_{2}$ for helium, however, and for this purpose an analysis of the processes occurring within the capillary is required. It is found that if the thermodynamic properties of the fluid in the capillary are not changed from those of bulk liquid, then Gorter's equation is correct. There seems, however, to be no reason to suppose that they may not be changed, so it is concluded that no direct information regarding ${\overline{S}}_{2}$ can be obtained from experiments on the thermomechanical effect. This last statement is considered in connection with some speculations concerning the state of liquid helium in a capillary and in a Rollin film.