# Solving Maxwell Relations Homework with Van der Waals Gas

• subzero0137
In summary, the conversation discusses finding the values for ##P + \left( \frac {∂U}{∂V} \right)_T## and ##\left( \frac {∂V}{∂T} \right)_P## using the Van der Waals equation and taking the differential of both sides. The resulting expressions are then plugged into the top equation to solve for CP - CV. The conversation also mentions difficulties in finding ##\left (\frac {∂V}{∂T} \right)_P##, but suggests using the product rule to solve for it.

## Homework Statement

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I'm stuck on part c of the attached problem:

## Homework Equations

$$C_P - C_V = \left[P + \left( \frac {∂U}{∂V} \right)_T \right]\left( \frac {∂V}{∂T} \right)_P$$

$$P + \left( \frac {∂U}{∂V} \right)_T = T \left( \frac {∂P}{∂T} \right)_V$$

$$\left(P + \frac {a}{V^2} \right)(V - b) = RT$$

## The Attempt at a Solution

I need to use the bottom two equations to find to find ##P + \left( \frac {∂U}{∂V} \right)_T ## and ##\left( \frac {∂V}{∂T} \right)_P## and plug these expressions in the top equation for CP - CV.

I've found $$\left (\frac {∂P}{∂T} \right)_V = \frac {R}{V - b}$$
$$∴ P + \left( \frac {∂U}{∂V} \right)_T = \frac {RT}{V - B}$$

But I'm having trouble finding ##\left (\frac {∂V}{∂T} \right)_P## because I can't seem to make V the subject of the Van der Waals gas expression.

#### Attachments

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Take the Van der Waal's equation and take a differential of both sides, using the product rule, (or just multiply out the left side), and treating ## P ## as a constant. You should get a couple of terms that are multiplied by ## dV ##. On the right side you will have ## R \, dT ##.

Last edited:
Take the Van der Waal's equation and take a differential of both sides, using the product rule, (or just multiply out the left side), and treating ## P ## as a constant. You should get a couple of terms that have are multiplied by ## dV ##. On the right side you will have ## R \, dT ##.

Oh I see. Thanks, I got it now :)

## 1. What are Maxwell relations?

Maxwell relations are a set of equations derived from the thermodynamic potentials that relate different properties of a system, such as temperature, pressure, volume, and entropy.

## 2. How can Maxwell relations be used to solve problems involving Van der Waals gas?

Maxwell relations can be used to solve problems involving Van der Waals gas by providing a way to relate the various thermodynamic properties of the gas, such as its pressure, volume, and temperature, to each other. This allows for the calculation of one property given the values of the others, making it a useful tool for solving problems involving Van der Waals gas.

## 3. What is the Van der Waals equation of state?

The Van der Waals equation of state is a modification of the ideal gas law that takes into account the size of gas molecules and the attractive forces between them. It is given by (P + a/V^2)(V - b) = RT, where P is the pressure, V is the volume, T is the temperature, a is a constant related to the intermolecular forces, and b is a constant related to the size of the gas molecules.

## 4. How does the Van der Waals equation of state improve upon the ideal gas law?

The Van der Waals equation of state improves upon the ideal gas law by taking into account the size of gas molecules and the attractive forces between them, which the ideal gas law does not consider. This makes it a more accurate representation of the behavior of real gases, particularly at high pressures and low temperatures.

## 5. How are Maxwell relations derived?

Maxwell relations are derived using the fundamental thermodynamic equations, such as the first and second laws of thermodynamics and the definitions of the thermodynamic potentials. They are based on the concept of exact differentials, which allows for the interchangeability of partial derivatives of state variables, leading to the derivation of the Maxwell relations.