# Derive Internal Energy from Thermodynamic Identity

• SalfordPhysics
In summary, we are asked to derive the internal energy (U) for a single molecule using the partition function (Z) and the thermodynamic identity. By applying the basic rules of differentiation, we can solve for U using the known values of S and F in the equation U = F + TS. This results in the simplified equation δU = -pδV + TδS.
SalfordPhysics

## Homework Statement

For a single molecule, derive the internal energy U = 3/2kBT
In terms of the partition function Z, F = -kBTlnZ
Where Z = V(aT)3/2

## Homework Equations

Thermodynamic identity: δF = -SδT - pδV
p = kBT/V
S = kB[ln(Z) + 3/2]

## The Attempt at a Solution

U = F + TS
δU = δF + δT.S + δS.T
= -SδT - pδV + δT.S + δS.T
δU = -pδV + TδS

(δS/δU)V = 1/T

However, don't have a variable U in S to differentiate with respect to.

Hint: don't forget the basic rules of differentiation, namely $\left(\frac{\partial}{\partial U}\right)_{V}S(Z) = \left(\frac{\partial Z}{\partial U}\right)_{V}\left(\frac{\partial}{\partial Z}\right)_{V}S(Z)$

I still don't see how I can solve it given that I don't have a term U to use.
The only thing I thought was solving p.dV and then substituting using U = -pdV but thing started getting messy.
Can't go for (dS/dU)V because again, no U.

Solved it, and I was just massively overcomplicating things. Just needed to use S and F known in U = F + TS

Last edited:

## 1. What is the thermodynamic identity?

The thermodynamic identity is a fundamental equation in thermodynamics that relates the internal energy, entropy, and pressure of a system. It 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.

## 2. How is the internal energy derived from the thermodynamic identity?

The internal energy can be derived from the thermodynamic identity by rearranging the equation and solving for the internal energy. This can be done by adding the work term to both sides and then integrating with respect to temperature.

## 3. What is the significance of deriving internal energy from the thermodynamic identity?

Deriving internal energy from the thermodynamic identity allows us to understand the relationship between energy, heat, and work in a system. It also allows us to calculate the internal energy of a system, which is an important thermodynamic property.

## 4. Can the thermodynamic identity be applied to all systems?

Yes, the thermodynamic identity can be applied to all systems, as long as they are thermodynamically closed. This means that no matter or energy can enter or leave the system, allowing for the conservation of energy.

## 5. How does the thermodynamic identity relate to the first law of thermodynamics?

The thermodynamic identity is essentially a mathematical representation of the first law of thermodynamics, which states that energy cannot be created or destroyed, only converted from one form to another. The thermodynamic identity shows how energy is conserved in a system by accounting for heat and work.

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