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Anna57
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- 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 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?
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 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?
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