# Calculating temperature from entropy

• SoggyBottoms
In summary, we are considering a system of N particles with two possible states, ground state \epsilon_0 and excited state \epsilon_1, that is thermally isolated with a fixed internal energy U. To calculate the temperature of this system, we need to first determine the multiplicity, which can be approximated using the Stirling approximation. From there, we can use the Boltzmann equation to calculate the temperature as a function of the energy difference between the two states and the change in multiplicity with the addition of an extra particle.
SoggyBottoms
Consider N >> 1 particles that can either be in groundstate $\epsilon_0$ or excited state $\epsilon_1$ and are thermally isolated, so the internal energy is fixed at $U = (N - n) \epsilon_0 + n \epsilon_1$. We want to calculate the temperature of this system.

This is how I attempt it: First calculate the multiplicity. For two particles, the possible microstates are -2, 0, 0, 2 and for 3 particles it's -3, -1, 1, 3, etc, so that the multiplicity becomes:
$$\Omega(n) = \frac{N!}{\frac{N + n}{2}! \frac{N - n}{2}!}$$
The entropy is, using the Stirling approximation, $S = k_B \ln{\Omega(n)} = k_B(N \ln N - \frac{N + n}{2} \ln{\frac{N + n}{2}} - \frac{N - n}{2} \ln{\frac{N - n}{2}})$.

The temperature we can calculate from $T = \left(\frac{\partial S}{\partial U}\right)^{-1}$. This is where I don't know how to continue (not sure I am on the right track in the first place).

The temperature can be calculated from the Boltzmann equation:T = \frac{2}{Nk_B}\frac{\epsilon_1-\epsilon_0}{\ln \left(\frac{\Omega(n)}{\Omega(n-1)}\right)}

## 1. What is entropy and how does it relate to temperature?

Entropy is a measure of the degree of disorder or randomness in a system. It is directly proportional to temperature, meaning that as temperature increases, so does entropy.

## 2. How is temperature calculated from entropy?

The formula for calculating temperature from entropy is: T = ΔU/ΔS, where T is temperature, ΔU is the change in internal energy, and ΔS is the change in entropy.

## 3. Can temperature be calculated from entropy for all systems?

No, the equation for calculating temperature from entropy only applies to reversible processes, meaning that the system must go through a series of small changes without any external influence.

## 4. What units are used for temperature and entropy in this calculation?

Temperature is typically measured in Kelvin (K), while entropy is measured in joules per Kelvin (J/K).

## 5. Is there a limit to how accurately temperature can be calculated from entropy?

Yes, due to the limitations of measurement and the complexity of systems, there is a limit to how accurately temperature can be calculated from entropy. This is why experimental data is often used to determine temperature rather than relying solely on calculations.

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