# Van der waal gas in isothermal free expansion

• mewmew
In summary, a gas obeying the Van Der Waal equation of state undergoes a free expansion from volume Vi to Vf at a constant temperature T. The change in entropy of the gas can be found by using the equation du=Tds-Pdv+mdn, and taking into account the additional term -N²a/V. This can be modified to find the amount of heat needed to prevent a change in temperature and turn it into an isothermal process. The change in entropy can also be calculated using the equation for S in terms of T and V, as it is a state variable.
mewmew

## Homework Statement

A gas obeys the Van Der Waal equation of state. The gas undergoes a free expansion from volume Vi to Vf at a constant temperature T. Find the change in entropy of the gas.

du=Tds-Pdv+mdn

## The Attempt at a Solution

I can solve the problem assuming du= 0 but this shouldn't be right. I can't seem to find out how to find out how much heat enters to keep the gas at a constant temperature. Everything shows that for a free expansion q=0, but here this is not the case.

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mewmew said:

## Homework Statement

A gas obeys the Van Der Waal equation of state. The gas undergoes a free expansion from volume Vi to Vf at a constant temperature T. Find the change in entropy of the gas.

du=Tds-Pdv+mdn

## The Attempt at a Solution

I can solve the problem assuming du= 0 but this shouldn't be right. I can't seem to find out how to find out how much heat enters to keep the gas at a constant temperature. Everything shows that for a free expansion q=0, but here this is not the case.

Take a look at this

This does not do your problem exactly, but it does give a derivation of the vdW energy as an additional term -N²a/V added to the ideal gas energy. It uses that result for a calculation of the change in temperature in a free expansion of a vdW gas from an initial volume to a final volume where V is so large that the addition term is neglected. If seems to me this could be easily modified to keep the additional term for any final volume. So you should be able to express a differential change in temperature in terms of a differential change in volume and relate that to the amount of heat needed to prevent that change in temperature and turn it into an isothermal process.

There is an equation for S in terms of T and V that may be just what you are looking for. If indeed S is a state variable, then the equation for S should be valid. Here is another site that seems to reach the same copnclusion. The difference in entropies at T2,V2 and T1,V1 is given as depending only on the initial and final states.

http://theory.phy.umist.ac.uk/~judith/stat_therm/node51.html

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## 1. What is a Van der Waal gas in isothermal free expansion?

A Van der Waal gas is a type of gas that follows the Van der Waals equation, which takes into account the intermolecular forces between gas particles. Isothermal free expansion refers to the process in which a gas expands without any change in temperature, resulting in a decrease in pressure.

## 2. How is the Van der Waal gas equation related to isothermal free expansion?

The Van der Waals equation describes the behavior of a gas undergoing isothermal free expansion by considering the volume of the gas and the intermolecular forces between particles. It is used to calculate the final pressure and volume of the gas after expansion.

## 3. What is the significance of isothermal free expansion in thermodynamics?

Isothermal free expansion is a fundamental process in thermodynamics that helps to understand the relationship between pressure, volume, and temperature of a gas. It also illustrates the concept of entropy, which is the measure of disorder in a system.

## 4. What factors affect the behavior of a Van der Waal gas in isothermal free expansion?

The behavior of a Van der Waal gas in isothermal free expansion is affected by the initial volume and pressure of the gas, as well as the strength of the intermolecular forces between particles. The temperature of the gas remains constant during this process.

## 5. How does isothermal free expansion differ from adiabatic expansion?

Isothermal free expansion and adiabatic expansion both involve the expansion of a gas, but the difference is in the temperature change. In isothermal free expansion, the temperature remains constant, while in adiabatic expansion, the gas is thermally insulated and the temperature may change. This results in different equations being used to calculate the final state of the gas.

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