Why Can Energy and Volume Changes Be Considered Separately in Thermodynamics?

[aaaa202]

dU = TdS - P dV

Is in my book derived by viewing a proces of changing volume and energy in two separate steps. First you add energy with volume fixed, then change the volume.
I'm just not sure that I understand why, you are allowed to do this. I know the changes are infinitesimal but why is it, that you are allowed to assume that the energy first changes, and then after that has happened the volume changes..

 

[Rap]

dU = TdS - P dV

Is in my book derived by viewing a proces of changing volume and energy in two separate steps. First you add energy with volume fixed, then change the volume.
I'm just not sure that I understand why, you are allowed to do this. I know the changes are infinitesimal but why is it, that you are allowed to assume that the energy first changes, and then after that has happened the volume changes..

You can view it that way, but you don't have to. You have an initial state and a final state that's very close to the initial state. You can jump from the initial to the final in one little step, two little steps (as above), three little steps, even a big loop that takes you far away from the initial state and then back to the final state. That's what the law is saying - if you start at the initial state and wind up at the final state (that is very close to the initial state), it doesn't matter what steps you took to get there, that equation will hold. (as long as you do it reversibly).

 

[aaaa202]

so for a gas that assumption would only hold, if it expands quasistatically?
I'm not quite sure what you mean by reversible to be honest.

 

[Hobold]

Reversibility is a tricky concept. A quasi-static process may not be reversible, but a reversible process must be quasi-static because you must be able to track every previous state you had to access in order to get to the desired one.

A process is reversible if you can control the heat your system exchanges with the exterior, that is, all forces in your system must be conservative. Therefore if you push a block which has friction with the ground, this process is not reversible because you cannot take the heat generated by friction and insert back in your block for it to "push itself".

 

[sardar]

The thermodynamic identity, also known as the first law of thermodynamics, is a fundamental equation in thermodynamics that relates changes in energy (dU), temperature (T), entropy (S), and pressure (P). It is derived from the principles of conservation of energy and the second law of thermodynamics.

In the context of changing volume and energy in two separate steps, the thermodynamic identity can be understood as follows: first, energy is added to the system while keeping the volume fixed, resulting in a change in internal energy (dU). This change in energy is then related to the change in entropy (TdS) and the work done by the system (PdV). This is a result of the fact that energy can be converted into work and vice versa, and the total energy of the system remains constant.

The reason why we can consider these changes in separate steps is because thermodynamic processes are typically considered to be reversible, meaning that they can be reversed without any loss of energy or increase in entropy. In this case, we can assume that the energy is added first, and then the volume changes, without affecting the overall outcome.

It is important to note that this is a simplification and in reality, changes in energy and volume may occur simultaneously. However, for small changes, the thermodynamic identity is a valid approximation and is widely used in thermodynamic calculations.

In summary, the thermodynamic identity is a fundamental equation that describes the relationship between energy, temperature, entropy, and pressure in a thermodynamic system. It is derived from the principles of conservation of energy and the second law of thermodynamics and is a valid approximation for reversible processes.