- #1

EE18

- 112

- 13

- Homework Statement
- I am working Callen Problem 3.4-13 which reads:

An impermeable, diathermal, and rigid partition divides a container into two subvolumes, each of volume ##V##. The subvolumes contam, respectively, one mole of H2 and three moles of Ne. The system is maintained at constant temperature ##T##. The partition is suddenly made permeable to H2, but not to Ne, and equilibrium is allowed to reestablish. Find the mole numbers and the pressures.

- Relevant Equations
- See below.

I am only interested in the initial equilibrium conditions, and I am struggling to convince myself whether that should correspond to the equality of chemical potentials for H2 or an equality of temperatures as well. My work is as below:

We take both gases as simple ideal (this is only relevant for later, and as mentioned no worries about this part). We could write ##dS## in terms of all of the possible ##dX_i## for extensive parameters ##X_i## which can be independently varied in a virtual process from the final constrained equilibrium, but it should be clear that the only possible parameters which can be varied are ##dU_A## and ##dN_{2,A}## (we let 1 denote Ne and 2 H\textsubscript{2} and ##A## and ##B## denote the two subvolumes) since ##dU_A = -dU_B## and ##dN_{2,A} = -dN_{2,B}## and the other parameters cannot vary. One further has, from conservation of energy, that in this constrained equilibrium the only possible virtual processes involve mass transfers so that we arrive at ##dU_A = \mu_{2,A}dN_{2,A}## (this is the chemical work noted in Chapter 1.8). Thus our equilibrium maximization condition is

$$0 = dS = \left(\frac{1}{T_A} - \frac{1}{T_B}\right)\mu_{2,A}dN_{2,A} + \left(\frac{\mu_{2,A}}{T_A} - \frac{\mu_{2,B}}{T_B}\right)dN_{2,A}$$

which, since ##dN_{2,A}## is arbitrary, implies that

$$ 0 =\left(\frac{1}{T_A} - \frac{1}{T_B}\right)\mu_{2,A} + \left(\frac{\mu_{2,A}}{T_A} - \frac{\mu_{2,B}}{T_B}\right) $$

But I can't go further than this. I can see that the equality of temperatures and chemical potentials is sufficient for this condition, but is it necessary?

We take both gases as simple ideal (this is only relevant for later, and as mentioned no worries about this part). We could write ##dS## in terms of all of the possible ##dX_i## for extensive parameters ##X_i## which can be independently varied in a virtual process from the final constrained equilibrium, but it should be clear that the only possible parameters which can be varied are ##dU_A## and ##dN_{2,A}## (we let 1 denote Ne and 2 H\textsubscript{2} and ##A## and ##B## denote the two subvolumes) since ##dU_A = -dU_B## and ##dN_{2,A} = -dN_{2,B}## and the other parameters cannot vary. One further has, from conservation of energy, that in this constrained equilibrium the only possible virtual processes involve mass transfers so that we arrive at ##dU_A = \mu_{2,A}dN_{2,A}## (this is the chemical work noted in Chapter 1.8). Thus our equilibrium maximization condition is

$$0 = dS = \left(\frac{1}{T_A} - \frac{1}{T_B}\right)\mu_{2,A}dN_{2,A} + \left(\frac{\mu_{2,A}}{T_A} - \frac{\mu_{2,B}}{T_B}\right)dN_{2,A}$$

which, since ##dN_{2,A}## is arbitrary, implies that

$$ 0 =\left(\frac{1}{T_A} - \frac{1}{T_B}\right)\mu_{2,A} + \left(\frac{\mu_{2,A}}{T_A} - \frac{\mu_{2,B}}{T_B}\right) $$

But I can't go further than this. I can see that the equality of temperatures and chemical potentials is sufficient for this condition, but is it necessary?

Last edited: