Calculating the work done during an isothermal expansion using integration

[bsmm11]

Homework Statement

In calculus, the work done when a gas expands from volume V₁ to volume V₂ is given by
W = ∫^V2_V1 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)

Homework Equations

Q = ΔU + W
PV = nRT

The Attempt at a Solution

W = [VP]^V2_V1 = 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.

 

[spacelike]

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.

 

[emailanmol]

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.