Solve Ideal Gas Problem Homework: Compute Work Done by Air

[jeff.berhow]

Homework Statement

Air that initially occupies 0.140m³ (V₁) at a gauge pressure of 103.0 kPa (p₁) is expanded isothermally to a pressure of 101.3 kPa (p₂) and then cooled at constant pressure until it reaches its initial volume. Compute the work done by the air.

Homework Equations

W_isothermal = nRTln(V₂/V₁)
W_isobaric = p₂ΔV = p₂(V₁ - V₂)
W_isothermal + W_isobaric = W_total

The Attempt at a Solution

We can find V₂ easily enough by multiplying p₁ and V₁ and dividing by p₂ because both p₁V₁ and p₂V₂ equal nRT. I can now fill in most of the formula:

nRTln(V₂/V₁) + p₂(V₁ - V₂) = W_total

Here, I thought I could then just back substitute nRT for either p₁V₁ or p₂V₂ and I'd be golden but the answer I'm getting is different from the back of the book (5.6 kJ). I'm getting .001 kJ with plugging these in:

W_total = (103kPa)(0.140m³)ln(0.140m³/.141m³) + (101.3 kPa)(.140m³ - .141m³)

Unfortunately, if I use the actual pressure, I still don't get the desired answer:

W_total = (204kPa)(0.140m³)ln(0.140m³/.141m³) + (202.3 kPa)(.140m³ - .141m³)

What am I fundamentally missing here? Thanks in advance.

*** ANSWER ***

They may be pulling a sneaky one on you. Only the first pressure given is specified as gauge pressure. Treat the second pressure as actual pressure. (That would make this a much more realistic problem!)

W_total = (204kPa)(0.140m³)ln(0.140m³/.141m³) + (101.3 kPa)(.140m³ - .141m³) = 5.6 kPa

 

[Doc Al]

For one thing, how does gauge pressure relate to absolute pressure?

 

[jeff.berhow]

Gauge pressure is the difference between the atmospheric pressure and the actual pressure in the object.

 

[Doc Al]

Gauge pressure is the difference between the atmospheric pressure and the actual pressure in the object.

Right. But in your calculation, you used the gauge pressure. (At least that's what you wrote.)

 

[jeff.berhow]

Right. But in your calculation, you used the gauge pressure. (At least that's what you wrote.)

You're absolutely right, Doc Al. I've updated my original post to reflect that. Thanks! :)

 

[Doc Al]

Air that initially occupies 0.140m³ (V₁) at a gauge pressure of 103.0 kPa (p₁) is expanded isothermally to a pressure of 101.3 kPa (p₂) and then cooled at constant pressure until it reaches its initial volume. Compute the work done by the air.

They may be pulling a sneaky one on you. Only the first pressure given is specified as gauge pressure. Treat the second pressure as actual pressure. (That would make this a much more realistic problem!)

 

[jeff.berhow]

That was it! Man, it feels like this is a reading comprehension problem more than an actual physics problem. I was really worried that I wasn't understanding something very fundamental about pressures, temperatures and volumes.

Thanks Doc Al!