# Mean energy and preassure inolving partition function

• rayman123
I understand how to calculate the mean energy and pressure of a gas using the partition function and Boltzmann's probability distribution.
rayman123

## Homework Statement

Can someone explain to me how this work out this formula

## Homework Equations

we are supposed to find the mean energy and preasure of a gas with given partition function

## The Attempt at a Solution

mean energy is given $$\overline{-}U=\sum_{r}E_{r}p_{r}$$
I know also that Boltzman's probability distribution is described by
$$p_{r}= \frac{e^{-\beta E_{r}}}{\sum_{r}e^{-\beta E_{r}}}$$

because the partition function is definied as $$z=\sum_{r}e^{-\beta E_{r}^}$$

so rewriting now the Boltzman's probablility distribution I get
$$p_{r}= \frac{e^{-\beta E_{r}}}{z}$$

## Homework Statement

now going back to the mean energy I can write
$$\overline{-}U=\frac{1}{z}\sum_{r}E_{r}e^{-\beta E_{r}$$

These are operations I do not understand. Could someone explain them step by step ?

$$\sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z$$

and the final one

$$U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}$$

rayman123 said:
These are operations I do not understand. Could someone explain them step by step ?
$$\sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z$$
The first part of the equation is how you find the (ensemble) average energy---you find the weighted average over all ensemble states. The second part is showing that you can rewrite this as the partial derivative with respect to beta; if you take the derivative of the Boltzmann factor WRT beta, the energy E gets pulled down
$$\frac{\partial}{\partial \beta}e^{-\beta E_{r}}= E_{r} e^{-\beta E_{r}}$$
In the expression using the partial derivative wrt beta, the sum over Boltzmann factors is the definition of the partition function. So you end up with the partial derivative of the partition function = the average energy.

rayman123 said:
and the final one
$$U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}$$
Again, this is just rewriting things with derivatives. Using the chain rule, if you take the derivative of lnz, you get a 1/z term out front.

'z' is some function of beta $$z = z(\beta)$$, so using the chain rule: $$\frac{\partial ln z(\beta)}{\partial \beta} = \frac{1}{z} \frac{\partial z(\beta)}{\partial \beta}$$

Does that make sense?

Last edited:
yes it does!:) thank you

## 1. What is the partition function and why is it important in calculating mean energy and pressure?

The partition function is a mathematical concept used in statistical mechanics to describe the distribution of energy among particles in a system. It is important in calculating mean energy and pressure because it allows us to determine the thermodynamic properties of a system by relating them to the microscopic properties of its constituents.

## 2. How is the partition function related to the Boltzmann distribution?

The partition function is directly related to the Boltzmann distribution, which describes the probability of a particle having a certain energy level. The Boltzmann distribution is derived from the partition function, making it a crucial component in understanding the energy distribution of a system.

## 3. Can you explain the difference between mean energy and pressure in terms of the partition function?

Mean energy is a measure of the average energy of a system, while pressure is a measure of the force exerted by particles in the system. The partition function is used to calculate both of these quantities, but they are fundamentally different in terms of what they represent in a system.

## 4. How does the number of particles in a system affect the partition function?

The number of particles in a system has a direct impact on the partition function. As the number of particles increases, the partition function also increases, indicating a higher probability of energy distribution among the particles. This relationship is important in understanding the behavior of large systems.

## 5. What other thermodynamic properties can be calculated using the partition function?

In addition to mean energy and pressure, the partition function can also be used to calculate other thermodynamic properties such as entropy, heat capacity, and chemical potential. It provides a powerful tool for understanding the behavior of complex systems and predicting their thermodynamic properties.

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