Undergrad Identification of changes in internal energy with work (in Callen's Thermodynamics)

[Anna57]

TL;DR

A question about the definition of internal energy and the identification of changes with adiabatic work as presented in Callen and similar literature.

Hello!

I have recently been reflecting on the formal structure of the second edition of Callen's Thermodynamics and an Introduction to Thermostatistics. Callen essentially postulates the existence of a function U, called the "internal energy", as a coordinate of thermodynamic systems. He doesn't explicitly say this, but it follows from the rest of the postulates that it must be defined for all equilibrium states, be continuously differentiable, additive over constituent subsystems, single-valued, and homogenous of the first order.

In section 1-7, titled "Measurability of the energy", Callen claims that given certain walls, called adiabatic walls, the mechanical work is a state function. In particular, he writes on p. 17:

The entire matter of controllability and measurability of the energy can be succinctly stated as follows: There exist walls, called adiabatic, with the property that the work done in taking an adiabatically enclosed system between two given states is determined entirely by the states, independent of all external conditions. The work done is the difference in the internal energy of the two states.

The particular part of this that irritates me somewhat is the conclusion that "The work done is the difference in the internal energy of the two states.". This is not shown anywhere and thus makes me wonder how one reaches this conclusion purely based of the postulated properties of the internal energy. I have an attempt to derive it below, but I am not sure how correct it is and there is a detail which is not entirely clear.

Let A, B and C be three equilibrium states of an adiabatically enclosed system under consideration accesible to each other in that order (A->B->C). For simplicity, let us assume that the system is simple and thus can be characterized completely by U, V and N in equilibrium.

Suppose that A, B and C have identical chemical composition and volume. Then, between each transformation, what must have changed is the internal energy. Clearly then, the changes in internal energy are functions of the adiabatic work,

$$\displaystyle \Delta U=f(x)$$

for some function f. Because the internal energy is a state function, it must be true that

$$\displaystyle f(W_{\text{ad.}}^{A\to B}+W_{\text{ad.}}^{B\to C})=f(W_{\text{ad.}}^{A\to B})+f(W_{\text{ad.}}^{B\to C})$$

This functional equation is satisfied by any linear map. However, because of the postulated continous differentiability of U, f must also be continuously differentiable (I think?), which leaves us with the solution below, for some real k

$$\displaystyle \Delta U^{X\to Y} = kW_{\text{ad.}}^{X\to Y}$$

A few questions about this:

(1) Is my reasoning correct?

(2). If it is, is there anything in the postulates of U that forces k=1, or is it merely a convenient choice? Clearly k=0 is unphysical, but other than that, are we technically free to pick any factor we would like?

 

[TSny]

In section 1-7, titled "Measurability of the energy", Callen claims that given certain walls, called adiabatic walls, the mechanical work is a state function. In particular, he writes on p. 17:

The particular part of this that irritates me somewhat is the conclusion that "The work done is the difference in the internal energy of the two states.". This is not shown anywhere and thus makes me wonder how one reaches this conclusion purely based of the postulated properties of the internal energy.

I think Callen’s conclusion follows from his discussions earlier in Chapter 1.

The change in the internal energy of a system during a process equals the net amount of energy transferred to the system during the process. This follows from the principle of conservation of energy.

For simple thermodynamic systems, work and heat are the two types of energy transfer. For an adiabatic process, there is no heat transfer. So, for an adiabatic process, the work done on the system equals $\Delta U$.

Since $\Delta U$ depends only on the initial and final states, the work done in an adiabatic process is determined solely by those states. Therefore, the work must be the same for all adiabatic processes that take the system from a specified initial state to a specified final state.

 

[Anna57]

I think Callen’s conclusion follows from his discussions earlier in Chapter 1.

The change in the internal energy of a system during a process equals the net amount of energy transferred to the system during the process. This follows from the principle of conservation of energy.

For simple thermodynamic systems, work and heat are the two types of energy transfer. For an adiabatic process, there is no heat transfer. So, for an adiabatic process, the work done on the system equals $\Delta U$.

Since $\Delta U$ depends only on the initial and final states, the work done in an adiabatic process is determined solely by those states. Therefore, the work must be the same for all adiabatic processes that take the system from a specified initial state to a specified final state.

Do you happen to have any resource discussing this fact? It is easy to show for macroscopic, mechanical energies that work is equivalent to the changes in energy. How is the same thing shown for microscopic particles? As far as I know, they don't obey Newtonian mechanics.

I understood this discussion in Callen as a mere motivation rather than a definition.

 

[TSny]

It is easy to show for macroscopic, mechanical energies that work is equivalent to the changes in energy. How is the same thing shown for microscopic particles? As far as I know, they don't obey Newtonian mechanics.

Even though we need quantum mechanics to describe atoms, we still assume that systems of atoms have well-defined energies and that the law of conservation of energy is valid. On page 17 of Callen, we read

When work is done on an adiabatically enclosed system, we can imagine it being performed by a mechanical contrivance, such as a compressed spring. The work, $W$, done by the spring on the system equals the loss of potential energy of the spring. Assuming conservation of energy, the loss of the spring’s potential energy must equal the gain in internal energy of the system, $\Delta U$. So, $\Delta U = W$.

Do you happen to have any resource discussing this fact?

You might find Chapter 2 in the little book The Elements of Classical Thermodynamics by A. B. Pippard to be a useful resource. In particular, pages 13-18.

Your mathematical argument in post #1 looks ok to me. (I don’t see why you assume that the states $A$, $B$, and $C$ have the same volume.) I think $k = 1$ follows from conservation of energy if you use the same units for $W$ and $U$.

 

[Anna57]

Even though we need quantum mechanics to describe atoms, we still assume that systems of atoms have well-defined energies and that the law of conservation of energy is valid. On page 17 of Callen, we read
View attachment 367978

When work is done on an adiabatically enclosed system, we can imagine it being performed by a mechanical contrivance, such as a compressed spring. The work, $W$, done by the spring on the system equals the loss of potential energy of the spring. Assuming conservation of energy, the loss of the spring’s potential energy must equal the gain in internal energy of the system, $\Delta U$. So, $\Delta U = W$.


You might find Chapter 2 in the little book The Elements of Classical Thermodynamics by A. B. Pippard to be a useful resource. In particular, pages 13-18.

Your mathematical argument in post #1 looks ok to me. (I don’t see why you assume that the states $A$, $B$, and $C$ have the same volume.) I think $k = 1$ follows from conservation of energy if you use the same units for $W$ and $U$.

Thank you for your reply!

I suppose the illustration with the spring makes sense: by Noether we know that the potential energy of the spring cannot just disappear. It stands to reason, then, that it must have been transferred into the system in some way (it is completely unreasonable that it would've gone anywhere else)

With regards to my mathematical argument, perhaps the restriction of constant volume was unnesecarily strong. I see now that I accidentally wrote "continuously differentiable", but I meant "continuous AND differentiable". Either way, the more I think about it, the more incorrect my argument seems.

The approach presented in Elements on Thermodynamics on p. 13-18, I had seen a few times before. My main concern with this way of defining the internal energy is this: how do we ensure that this quantity we define as differences in adiabatic work with respect to some fiducial state has all the desired properties? Can we prove things such as differentiability, continuity, etc.?

 

[TSny]

The approach presented in Elements on Thermodynamics on p. 13-18, I had seen a few times before. My main concern with this way of defining the internal energy is this: how do we ensure that this quantity we define as differences in adiabatic work with respect to some fiducial state has all the desired properties? Can we prove things such as differentiability, continuity, etc.?

I'm not familiar with any treatments of thermodynamics that prove differentiability of $U$. I think physicists tend not to worry too much about it. They just assume differentiability. For example, regarding entropy, Callen gives on page 28:

So, Callen postulates differentiability of $S$.