# Understand the Thermodynamic Identity: Is This Correct?

• I
• Sebas4
In summary, the Thermodynamic Identity is an equation that relates the change in internal energy of a system to the change in entropy and temperature, assuming constant volume and number of particles. It can be used to calculate the increase in energy in a system when heat is added, as long as the process is reversible or involves two closely neighboring thermodynamic equilibrium states.
Sebas4
I have a question about the Thermodynamic Identity.
The Thermodynamic Identity is given by
$$dU = TdS - PdV + \mu dN$$.
We assume that the volume $V$ and that the number of particles $N$ is constant.
Thus the Thermodynamic Identity becomes
$$dU = TdS$$.
Assume that we add heat to the system (we see that $dU = dQ$ because $dQ = TdS$ and the work done is 0, because $dV=0$).
We see that the entropy and the temperature of the system increase.
The increase of energy in the system is given by
$$\Delta U = \int TdS$$,
with $T$ the temperature of the system (which is not constant) and $dS$ the change in entropy of the system.
Is this correct?

I am not trying to calculate anything. I just want to know if this is correct or not.

- Sebas4.

Last edited:
If the heat is added irreversibly (i.e., very rapidly), the temperature of the system is not uniform spatially (and neither is the internal energy per unit mass and the entropy per unit mass). How would you use this equation under those circumstances?

Not, because the temperature of the system is not uniform (or equal). Is it possible to add heat in such a way that the process in quasistatic? In real life it is not possible but theoretically? Is the equation then valid?

Maybe I have to change my question. My question is what's is the use of the thermodynamic identity?

Sebas4 said:
Not, because the temperature of the system is not uniform (or equal). Is it possible to add heat in such a way that the process in quasistatic? In real life it is not possible but theoretically? Is the equation then valid?
Yes, in the reversible limit, the equation is valid.
Sebas4 said:
Maybe I have to change my question. My question is what's is the use of the thermodynamic identity?
The equation is valid in the reversible limit, and is also valid for two closely neighboring (differentially separated) thermodynamic equilibrium states, irrespective of how tortuous and irreversible the path between these states had been, provided only that they are differentially separated and each at thermodynamic equilibrium. Of course, a reversible path is a continuous sequence of thermodynamic equilibrium states.

## 1. What is the thermodynamic identity?

The thermodynamic identity is a fundamental equation in thermodynamics that relates the internal energy of a system to its temperature, pressure, and volume. It is also known as the first law of thermodynamics.

## 2. How is the thermodynamic identity derived?

The thermodynamic identity is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. By rearranging this equation, we can obtain the thermodynamic identity.

## 3. What is the significance of the thermodynamic identity?

The thermodynamic identity is significant because it allows us to understand and predict the behavior of thermodynamic systems. It also serves as the basis for many other thermodynamic equations and principles.

## 4. Is the thermodynamic identity always correct?

Yes, the thermodynamic identity is always correct as it is derived from the first law of thermodynamics, which is a fundamental law of nature. However, it may not always be applicable in certain situations or when certain assumptions are not met.

## 5. How is the thermodynamic identity used in practical applications?

The thermodynamic identity is used in various practical applications, such as in the design of heat engines, refrigeration systems, and power plants. It is also used in chemical thermodynamics to understand and predict the behavior of chemical reactions and processes.

• Thermodynamics
Replies
31
Views
2K
• Thermodynamics
Replies
4
Views
1K
• Biology and Chemistry Homework Help
Replies
0
Views
84
• Thermodynamics
Replies
3
Views
1K
• Thermodynamics
Replies
13
Views
2K
• Biology and Chemistry Homework Help
Replies
13
Views
200
• Thermodynamics
Replies
11
Views
509
• Thermodynamics
Replies
12
Views
2K
• Thermodynamics
Replies
10
Views
2K
• Thermodynamics
Replies
1
Views
961