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CAF123

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## Homework Statement

Consider a frictionless piston inside a cylinder, both with a cross sectional area A.

1) By considering the work done by the gas on the piston and the work dW required to move the piston a distance dx, show that the work done by the gas is dW = -pdV.

2) 1 L of an ideal gas is at a pressure of 5 atm. If it then expands at constant temperature against a frictionless piston until It's pressure falls back to atmospheric pressure, calculate I) the final volume of gas, 2) the work done on (or is it by?) the gas?

## The Attempt at a Solution

1) The force of the gas on the piston is PA. Then W = Fdx = PA dx = PdV. This is straightforward but I have a positive work. Is there a mistake in the question? dx and F are in same direction so by definition, this is positive work.

2) I) 5L ii) So dV = 4 L . The work done will be by the gas since it is the gas that is expanding and the piston does not act to compress. I don't think I can use the above relation because the pressure is not constant. I will try an integral. Is this the right way to proceed?

EDIT: The integral I have is $$W = nRT\int_{V_o}^{V_f} \frac{dV}{V} = nRT[ln(V_f) - ln(V_o)] = nRT ln \left(\frac{V_f}{V_o}\right)$$ I would have then an answer of ##nRT ln(5)##, since I don't know n or T and R is a constant.

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