# Thermodynamics without partition function

• cryptist
In summary, the conversation discussed ways to derive entropy and free energy without using the partition function in statistical mechanics. The Helmoholtz free energy and the heat capacity were mentioned as tools to calculate these quantities. However, it was noted that extracting the entropy from the heat capacity can be complicated. Ultimately, the speaker found the answer to their question and the conversation came to a close.
cryptist
Is there a way to derive entropy or free energy without using partition function?

Yes.

If you want a more complete answer, you'll have to post a more detailed question.

Thermodynamics came before statistical mechanics.

As you all know, N=∑ni, U=∑εi, F=Nμ-kT∑Z and S=(U-F)/T

Here, I do not want to use partition function Z. How do I write F and S then?

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The Helmoholtz free energy is defined as
$$F \equiv U - TS$$
Entropy you can get from the heat capacity:
$$C_V \equiv T \left( \frac{\partial S}{\partial T} \right)_V$$

Let me specify the problem. I am using grand canonical ensemble and I am in Fermi-Dirac statistics.

Considering these, entropy is written as S=k[lnZ+β(E-μN)]. How do I write S, without using Z?

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DrClaude said:
The Helmoholtz free energy is defined as
$$F \equiv U - TS$$
Entropy you can get from the heat capacity:
$$C_V \equiv T \left( \frac{\partial S}{\partial T} \right)_V$$

Heat capacity is a derived quantity. It is really complicated to extract S from Cv.

Jorriss said:
Thermodynamics came before statistical mechanics.

Ok. Then, how do I derive S without using partition function (a statistical mechanics tool)? Btw, by deriving S, I mean I'll calculate the entropy of a system over momentum states, I am not talking about S=klnΩ which apparently does not include partition function.

cryptist said:

It would be nice to share with us the answer you found

## 1. What is thermodynamics without partition function?

Thermodynamics without partition function is a simplified approach to studying thermodynamic systems without using the partition function, which is a mathematical tool used to calculate the thermodynamic properties of a system. This approach is often used in introductory thermodynamics courses to help students understand the basic principles of thermodynamics.

## 2. How is thermodynamics without partition function different from traditional thermodynamics?

In traditional thermodynamics, the partition function is used to calculate the thermodynamic properties of a system, such as energy, entropy, and free energy. In contrast, thermodynamics without partition function focuses on the qualitative understanding of thermodynamic concepts without relying on complex mathematical calculations.

## 3. What are the advantages of using thermodynamics without partition function?

One advantage of using thermodynamics without partition function is that it simplifies the study of thermodynamics and allows for a more intuitive understanding of the principles. It also eliminates the need for complex mathematical calculations, making it more accessible to beginners and non-mathematical scientists.

## 4. Are there any limitations to using thermodynamics without partition function?

Yes, thermodynamics without partition function is a simplified approach and therefore has limitations. It cannot be used to make precise quantitative predictions about a system's behavior, and it may not be applicable to more complex thermodynamic systems.

## 5. Is thermodynamics without partition function used in real-world applications?

No, thermodynamics without partition function is not used in real-world applications. It is mainly used as a teaching tool to introduce students to the basic concepts of thermodynamics before they move on to more advanced methods and applications.

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