# Gibbs' theorem and partial molar volume

• kayan
In summary, the chemical engineering text by Smith, VanNess, and Abbott discusses partial molar volume and states that Gibbs theorem applies to all partial molar properties except for volume. This is because for other properties, such as entropy, they are evaluated at T, Pi (partial pressure of i in the mixture), while volume is evaluated at T, P (total mixture pressure). This raises the question of how other properties, such as U, H, S, A, and G, still satisfy Gibbs theorem. The answer lies in the fact that these properties involve entropy and its relation to the sum of molar entropies of pure constituents at the same temperature and partial pressure in the mixture. While this may not be fully explained in
kayan
In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a mixture, we evaluate it at a T, P (total mixture pressure), but for other partial molar properties (like entropy), we evaluate them at T, Pi (partial pressure of i in the mixture).

There's a very similar thread on physics stack exchange, but I don't find a completely satisfying answer there (an neither did the OP): https://physics.stackexchange.com/questions/502788/gibbs-theorem-and-partial-molar-volume.

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What kind of answer were you looking for?

In the case of U and H, for an ideal gas mixture, these are independent of pressure, so the distinction "at the same partial" pressure doesn't come in. So these are just thrown in with the other properties that involve entropy.

S, A, and G all involve entropy S in their definitions. You can show that the reversible heat required to isothermally separate an ideal gas mixture into its pure constituents at their partial pressure in the mixture is zero. So the entropy of the mixture must be equal to the sum of the molar entropies of the pure constituents at the same temperature and pressure equal to their partial pressures in the mixture multiplied by the number of moles of each species. This means that S satisfies Gibbs theorem. And, since A and G involve U and H, respectively, and TS, if S satisfies Gibbs theorem, so must they.

So the real question should be "if V does not satisfy Gibbs theorem, how come U, H, S, A, and G do?"

Framing the question is definitely part of the problem, because since V, U, & H for an IG can all be evaluated at the total P, then Gibbs theorem seems to be the exception instead of the rule, with the exception being S (and anything related to S). I don't know what kind of answer I was looking for, just perhaps a better explanation of the rule instead of just accepting it by fiat, which is how any source that I've found has justified it.

It seemed like an issue that deserves more space in the textbooks than has been given to it. Your entropy explanation is something that makes sense, so I'm going to think about it for a bit and see if that resolves my concerns.

Don't be too hard on Smith and Van Ness. I think it's a really good book. Another great thermo book is Fundamentals of Engineering Thermodynamics by Moran et al. But the emphasis in not so much on ChE interests, such as solution thermodynamics.

Not meaning to bash Smith and VN at all. In fact, it's one of the only thermo books I've read that discusses this topic in any degree. I just wished they would've explained more since they are my only source on the subject!

## 1. What is Gibbs' theorem?

Gibbs' theorem, also known as the Gibbs-Duhem equation, is a fundamental principle in thermodynamics that relates the changes in the chemical potential of a mixture to its composition and temperature. It states that the sum of the partial molar volumes of all components in a mixture is equal to the total volume of the mixture.

## 2. How is Gibbs' theorem used in chemistry?

Gibbs' theorem is used to understand and predict the behavior of mixtures, particularly in the fields of physical chemistry and chemical engineering. It is often used to calculate the change in volume of a solution when a solute is added, and to determine the equilibrium conditions of a reaction.

## 3. What is partial molar volume?

Partial molar volume is the change in volume of a solution when one mole of a particular component is added while keeping the amounts of all other components constant. It is a measure of the change in volume due to the addition of a specific component, and is often used to characterize the interactions between different components in a mixture.

## 4. How is partial molar volume related to the properties of a solution?

Partial molar volume is related to the properties of a solution through its effect on the chemical potential of the components in the mixture. The partial molar volume of a component can affect the solubility, vapor pressure, and other properties of a solution. It is also used to calculate the excess volume of a mixture, which is a measure of the deviation from ideal behavior.

## 5. Can Gibbs' theorem be applied to non-ideal solutions?

Yes, Gibbs' theorem can be applied to non-ideal solutions as long as the components in the mixture are in thermodynamic equilibrium. However, the values of partial molar volumes may deviate from ideal behavior in non-ideal solutions, and additional equations or models may be needed to accurately predict the behavior of the mixture.

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