# Thermodynamics Finding a fundamental equation.

• corr0105
In summary, the question is about finding the fundamental function Z for natural variables (T,p,μ) in thermodynamics, specifically in relation to biological cells that regulate chemical potentials instead of number of molecules. The solution involves applying a Legendre transform, which allows for the use of μ as a natural variable instead of N. This information helped the individual figure out the solution to the problem.
corr0105
Thermodynamics! Finding a fundamental equation.

This is the question:
"While he Gibbs free energy G is the fundamental function of the atral variables (T,p,N), (T=temperature, p=pressure, N=number of molecules), growing biological cells often regulte not the numbers of molecules N, but the chemical potentials μ. That is, they control concentrations. What is the fundamental function Z of natural variables (T,p,μ)?

I know a few equations that deal with Gibbs free energy:
G=H-TS
G=$$\sum$$μN

Basically, I have no idea where to start with this problem. If anyone can give me a push in the right direction that would be much appreciated!

You need to apply a Legendre transform (so check out what this is). An example of of Legendre transform: we use G instead of H because we wish to use T as a natural variable instead of S. Now you've learned that you'd also like to use μ instead of N. This should be enough to get you started.

Thanks so much, I looked up how to use Legendre transforms for this type of problem and eventually figured it out (not to mention learned something!). Thanks for your help!

## 1. What is thermodynamics?

Thermodynamics is the branch of science that deals with the relationships between heat, energy, and work. It studies how energy is transferred and transformed from one form to another.

## 2. What is a fundamental equation in thermodynamics?

A fundamental equation in thermodynamics is an equation that describes the fundamental principles of the system and can be used to calculate various thermodynamic properties, such as temperature, pressure, and entropy.

## 3. How is a fundamental equation derived?

A fundamental equation is derived from the laws of thermodynamics, which are based on the conservation of energy and the concept of entropy. It is then refined and adapted to fit specific systems or processes.

## 4. What are the variables in a fundamental equation?

The variables in a fundamental equation depend on the specific system being studied, but they typically include temperature, pressure, volume, and entropy. Other variables may also be included depending on the complexity of the system.

## 5. How is a fundamental equation used in practical applications?

A fundamental equation is used in practical applications to predict and calculate thermodynamic properties of a system, such as the efficiency of a heat engine or the behavior of a chemical reaction. It serves as a basis for understanding and analyzing various processes and phenomena in the natural world.

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