# Calculating the work done during an isothermal expansion using integration

• bsmm11
In summary: Use ideal gas equationP=nRT/V, now you can put this value in your integral and integrate.In summary, the work done when a gas expands from volume V1 to volume V2 is given by W = ∫V2V1 P dV. Using this expression, it can be shown that the work done by n moles of gas at temperature T during an isothermal expansion from volume V1 to V2 is W = nRT ln(V2/V1). This is found by using the ideal gas law, PV = nRT, and integrating the expression PdV with T being held constant.
bsmm11

## Homework Statement

In calculus, the work done when a gas expands from volume V1 to volume V2 is given by
W = ∫V2V1 P dV
Use this expression to show that the work done by n moles of gas at temperature T during an isothermal expansion from volume V1 to V2 is
W = nRT ln(V2/V1)

Q = ΔU + W
PV = nRT

## The Attempt at a Solution

W = [VP]V2V1 = PV2 - PV1 = PΔV
But I think it should be ΔPΔV since this is an isothermal expansion. W = PΔV is for isobaric since P is constant.
Then I can't even guess where the ln comes from.

You got P as a constant because you treated it like one when you took the integral.

But if you look at the ideal gas law you can see that pressure is a function of volume. So then you can put that expression into the integral and n, R, and T are constants, then integrate.

Hey,

Unfortunately You have got it wrong.

See work is defined as
dW =PdV , where P is external pressure and V is small volume change.

This comes from the fact that dW=Force * displacement

dW=(External)Pressure*Area*displacement

However area * displacement is change in volume so
dW=PdV

You have to integrate this expresion to get the value of work.

Now in isothermal reversible conditions , you have to find work done by system which is a GAS
In such cases pressure external =pressure of the gas.

Remember, Ideal gas Equation.?

How will you integrate PdV now with T being constant.

## What is an isothermal expansion?

An isothermal expansion is a thermodynamic process in which a gas expands while being kept at a constant temperature. This means that the internal energy of the gas remains constant and all the heat added to the gas is used to do work.

## What is the formula for calculating the work done during an isothermal expansion using integration?

The formula for calculating the work done during an isothermal expansion using integration is W = ∫PdV, where W is the work done, P is the pressure and dV is the change in volume.

## Why is integration used to calculate the work done during an isothermal expansion?

Integration is used because the pressure and volume of the gas are not constant during the expansion. Integration allows us to find the total work done by taking into account the changing pressure and volume over the entire expansion process.

## What are the units for work done during an isothermal expansion?

The units for work done during an isothermal expansion are joules (J) in the SI system and calories (cal) in the CGS system.

## Can the work done during an isothermal expansion be negative?

Yes, the work done during an isothermal expansion can be negative. This means that the gas is doing work on its surroundings instead of the surroundings doing work on the gas. This can happen when the external pressure on the gas is greater than the internal pressure, causing the gas to compress instead of expand.

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