1st Law of Thermo, work in a piston.

[twmggc]

Homework Statement

Air is expanded in a piston-cylinder arrangement at a constant P of 200kPa from a volume of 0.1 m³ to a volume of 0.3m³. Then the temperature is held constant during an expansion to a volume of 0.5m³. Predict the total work done in the air.

Homework Equations

W = ∫PdV , PV = nRT

The Attempt at a Solution

For the first part of the expansion I used:
W = ∫PdV = P∫dV = P(V₂ - V₁) = 200,000Pa ( 0.3-0.1)m³
and got W = 40,000J.

The next expansion is what is confusing me.
Since Pressure is no longer constant I need to leave it in the W = ∫PdV equation.
So, I transform this equation using the ideal gas eqn. PV = nRT and get:
W = nRT∫dV/V = nRT*ln(V3/V2)
But, without knowing the n or the T how do I get the work?

Thanks!

 

[TSny]

Trick: Replace the nRT in your expression for W with some other expression (using the ideal gas law).

 

[twmggc]

Since Temperature is constant for this part I can assume P₂V₂=P₃V₃ and find P₃ and then replace nRT in my work equation with P₃V₃ (PV = nRT).
After my computations I got 30,649.54J which sounds within reason.

Was this what you were thinking?

 

[TSny]

Yes. But there's no need to find P₃. You can use P₂ and V₂ instead of P₃ and V₃.

 

[Nickie]

I would respond by saying that the first law of thermodynamics states that energy cannot be created or destroyed, only transferred or converted from one form to another. In this case, the air is expanding and doing work on the piston, causing it to move. The work done by the air is equal to the area under the pressure-volume curve, which can be calculated using the equation W = ∫PdV.

For the first part of the expansion, since the pressure is constant, we can simply use the equation W = PΔV to calculate the work done. However, for the second part of the expansion where the temperature is held constant, we cannot use this equation since the pressure is no longer constant. Instead, we can use the ideal gas law, PV = nRT, to calculate the work done.

To do this, we first need to find the number of moles (n) of air present. This can be done using the ideal gas law, n = PV/RT. Since the temperature is held constant, we can use the initial pressure and volume to calculate n. Plugging this value of n into the equation W = nRT*ln(V3/V2), we can then calculate the work done during this part of the expansion.

In summary, the total work done by the air is the sum of the work done during the first part of the expansion (W = PΔV) and the work done during the second part of the expansion (W = nRT*ln(V3/V2)).