# Thermo: isothermal, reversible expansion of ideal gas

#### Ryaners

1. Homework Statement
Two moles of a monatomic ideal gas are at a temperature of 300K. The gas expands reversibly and isothermally to twice its original volume. Calculate the work done by the gas, the heat supplied and the change in internal energy.
So:
T = 300K; ΔT = 0
n = 2; R = 8.314 J K-1 mol-1
V2 = 2V1
Reversible process ⇒ work is path independent.

2. Homework Equations
PV = nRT
W = -PdV
U = W + Q

3. The Attempt at a Solution
PV = nRT ⇒ P = nRT / V
dW = -PdV ⇒ W = - ∫ PdV = - nRT ∫ [dV / V] = nRT [ln |V2 - V1|] = nRT ln V1

This isn't very helpful as I have no actual figure for what V1 is.

I've tried to come up with something based on how the work is path independent (so W = P2V2 - P1V1 = V1[P2 - P1]) but I've just confused myself, as ΔT = 0 means that P1V1 = P2V2 = nRT, so that would mean that W = 0 which isn't right. I must be going wrong with my assumptions somewhere - any pointers would be much appreciated!

Edit: I'm confident that once I get past the first stumbling block & calculate the work done I can fire ahead & calculate Q and ΔU. Thanks in advance!

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#### Chestermiller

Mentor
You integrated incorrectly. Try again.

#### Ryaners

You integrated incorrectly. Try again.
Aha! I see what you mean & I get the right answer when evaluating the integral like this:
nRT [ln |V2 - V1|] = nRT [ln |V2 / V1| ] = nRT [ln | 2V1 / V1| ] = nRT ln

Thanks for the nudge in the right direction. May I ask: is it ever correct to write
nRT [ln |V2 - V1|]
or should it be
nRT [ln |V2| - ln |V1|]
when substituting end points in an integral of this form? Are these two expressions equivalent? (I'm now thinking they're not..?)

#### Chestermiller

Mentor
Aha! I see what you mean & I get the right answer when evaluating the integral like this:
nRT [ln |V2 - V1|] = nRT [ln |V2 / V1| ] = nRT [ln | 2V1 / V1| ] = nRT ln

Thanks for the nudge in the right direction. May I ask: is it ever correct to write
nRT [ln |V2 - V1|]
or should it be
nRT [ln |V2| - ln |V1|]
when substituting end points in an integral of this form? Are these two expressions equivalent? (I'm now thinking they're not..?)
They're not equivalent. You need to review some of the math related to this. When you evaluate results of a definite integral at the limits of integration, you write $[f(x)]_{x=a}^{x=b}=f(b)-f(a)$, not $f(b-a)$. Also, the argument of a combined natural log term should have no units.

#### Ryaners

They're not equivalent. You need to review some of the math related to this. When you evaluate results of a definite integral at the limits of integration, you write $[f(x)]_{x=a}^{x=b}=f(b)-f(a)$, not $f(b-a)$. Also, the argument of a combined natural log term should have no units.
Thanks, that makes it very clear. It's more that I have some uncertainty around logarithms, which I need to revise / practice again; i.e. ln(a) - ln(b) ≠ ln(a-b), which is obvious to me now but was the mistake I made yesterday without questioning.

Appreciate the help, thank you!

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