# Thermodynamic equilibrium for systems only open to particle exchange

• EE18
In summary: However, if you want to pursue this further, maybe you could look at a potential energy equation?Off hand, it is not clear to me the...relation between the chemical potential and the temperature. However, if you want to pursue this further, maybe you could look at a potential energy equation?
EE18
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?

Last edited:
Isn't the temperature T equal for the two chambers and constant over the process? Isn't the condition for equilibrium of H2 equal chemical potential in the two chambers? What is the equation for the chemical potential of an ideal gas component in a mixture?

Lord Jestocost, vanhees71 and Greg Bernhardt
Chestermiller said:
Isn't the temperature T equal for the two chambers and constant over the process? Isn't the condition for equilibrium of H2 equal chemical potential in the two chambers? What is the equation for the chemical potential of an ideal gas component in a mixture?
Oy, I completely skipped over the part about each subvolume being at ##T##, you are right. Can you comment incidentally on what would occur if not (as i thought was the case)? How does the condition I've arrived imply the equality of temperatures in that case?

EE18 said:
Oy, I completely skipped over the part about each subvolume being at ##T##, you are right. Can you comment incidentally on what would occur if not (as i thought was the case)? How does the condition I've arrived imply the equality of temperatures in that case?

Chestermiller said:
Isn't the temperature T equal for the two chambers and constant over the process?
Yes, equal initially.. It's actually not clear to me why it must necessarily remain so throughout the process (I know that ##U## is only a funciton of ##T## in an ideal gas, but there's nothing here which makes it clear a a priori that the temperatures in the subvolumes can't change as there's mass transfer, so perhaps my initial question stands)? Actually I see that the system is maintained at ##T##, so this is moot. But if it were an isolated system it would be interesting to think about.

Isn't the condition for equilibrium of H2 equal chemical potential in the two chambers?
I am trying to prove the relevant equilibrium. I agree that this is certainly the case if the connecting wall is not adiabatic. As I've shown here, it's not as clear if not. I think this is a similar sort of problem as Callen's adiabatic piston.

What is the equation for the chemical potential of an ideal gas component in a mixture?
I quote from an earlier problem I worked: Using the definition of the partial pressure ##P_j## from Problem 3.4.11, we can change the first term to be of the form below:
$$\mu_j := RT\ln \frac{P_jv_0}{RT} + f(T).$$

EE18 said:
Isn't the temperature T equal for the two chambers and constant over the process?
Yes, equal initially.. It's actually not clear to me why it must necessarily remain so throughout the process (I know that ##U## is only a funciton of ##T## in an ideal gas, but there's nothing here which makes it clear a a priori that the temperatures in the subvolumes can't change as there's mass transfer, so perhaps my initial question stands)? Actually I see that the system is maintained at ##T##, so this is moot. But if it were an isolated system it would be interesting to think about.

Isn't the condition for equilibrium of H2 equal chemical potential in the two chambers?
I am trying to prove the relevant equilibrium. I agree that this is certainly the case if the connecting wall is not adiabatic. As I've shown here, it's not as clear if not. I think this is a similar sort of problem as Callen's adiabatic piston.
Off hand, it is not clear to me the conditions for equilibrium if the partition is adiabatic rather than diathermal. But clearly, as you correctly deduced, for diathermal, the condition is equal temperatures.
EE18 said:
What is the equation for the chemical potential of an ideal gas component in a mixture?
I quote from an earlier problem I worked: Using the definition of the partial pressure ##P_j## from Problem 3.4.11, we can change the first term to be of the form below:
$$\mu_j := RT\ln \frac{P_jv_0}{RT} + f(T).$$
The equation is $$\mu_j=\mu_j^0(T,P^0)+RT\ln{(p_j/P^0)}$$where ##P_0## is a reference pressure (typically 1 bar), ##\mu_j^0(T,P^0)## is the molar free energy of pure species j at T and P^0, and ##p_j## is the partial pressure of species j in the mixture (or pure). The condition of equilibrium is equal chemical potentials in the two chambers for H2, or, equivalently, equal partial pressures. So the pressure of H2 in the chamber with pure H2 must be equal to the partial pressure of the H2 in the chamber with the mixture of Ne and H2.

EE18

## What is thermodynamic equilibrium in the context of systems open to particle exchange?

Thermodynamic equilibrium in systems open to particle exchange refers to a state where the system's macroscopic properties, such as temperature, pressure, and chemical potential, do not change over time. In this state, the rate of particle exchange between the system and its surroundings is balanced, resulting in no net change in the number of particles within the system.

## How is chemical potential defined for systems open to particle exchange?

Chemical potential is a measure of the change in a system's Gibbs free energy when an additional particle is introduced, keeping temperature and pressure constant. For systems open to particle exchange, the chemical potential must be equal between the system and its surroundings at equilibrium. This ensures that there is no net flow of particles.

## What conditions must be met for a system to be in thermodynamic equilibrium?

For a system to be in thermodynamic equilibrium while open to particle exchange, three conditions must be met: thermal equilibrium (uniform temperature), mechanical equilibrium (uniform pressure), and chemical equilibrium (uniform chemical potential). These conditions ensure that there are no gradients driving the flow of heat, work, or particles.

## How does the grand canonical ensemble relate to systems open to particle exchange?

The grand canonical ensemble is a statistical mechanics framework used to describe systems open to particle exchange. It allows for fluctuations in particle number and energy, characterized by the temperature, volume, and chemical potential. This ensemble is particularly useful for studying systems where the number of particles can vary, such as in chemical reactions or open quantum systems.

## What role does entropy play in thermodynamic equilibrium for open systems?

Entropy, a measure of disorder or randomness, tends to increase in any spontaneous process. For an open system at thermodynamic equilibrium, the entropy is maximized given the constraints of constant temperature, pressure, and chemical potential. This maximization of entropy ensures that the system is in its most probable state, with no net flow of particles or energy.

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