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CAF123
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Homework Statement
A metal container of volume ##V## and with diathermal walls contains ##n## moles of an ideal gas at high pressure. The gas is allowed to leak out slowly from the container through a small valve to the atmosphere at a pressure ##P_0##. The process occurs isothermally at the temperature of the surroundings. Show that the work done by the gas against the surrounding atmosphere is ##W = P_0 (nv_o - V)##, where ##v_o## is the molar volume of the gas at atmospheric pressure and temperature.
Homework Equations
Work done by gas: ##W = \int_{V_i}^{V_f} P\,dV##
The Attempt at a Solution
I think I have the answer, but I have one question and I don't get a physical implication that the equation suggests.
Reasoning for answer: Initially volume occupied by gas: V. After all gas released, the volume occupied by n moles is nvo. The gas does work against surroundings so ##W = \int_V^{nv_o} P_0 dV ## gives the answer.
However, one small point: Why can I say the volume occupied by the gas at the end when it is all released is ##nv_o##? This is only true under STP conditions and it is given that the pressure of the atmosphere is P0 which need not be atmospheric pressure. We don't know anything about the temperature either.
Physical implication: The equation appears to be independent of ##P## inside the container. But I think, via the ideal gas law, I can write ##V = \frac{nRT}{P}## and so the equation becomes $$W = nP_0\left(v_o - \frac{RT}{P}\right) $$ but I don't think this makes sense:
As P gets larger, RT/P tends to zero and so this tends to make W bigger. I would expect that W get smaller.
Many thanks.