InterviewSolution
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InterviewSolution
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Demonstrate that the process in which the work performed by an ideal gas is alphaortional to that corresponding increment of its internal energy is decribed by the equation `p V^n - const`, where `n` is a constant. |
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Answer» According to the problem : `A alpha U` or `dA = aU` (where `a` is alphaortionality constant) or, `pdV = (a v R dT)/(gamma - 1)`…(1) ltrgt From ideal gas law, `pV = v R T`, on differentiating `pdV + Vdp = v RdT`…(2) Thus from (1) and (2) `pdV = (a)/(gamma - 1) (pdV + Vdp)` or, `pdV ((a)/(gamma - 1) - 1) + (a)/(gamma - 1) V dp = 0` or, `pdV(k - 1) + kVdp = 0`(where `k = (a)/(gamma - 1) = "another constant")` or, `pdV (k - 1)/(k) + Vdp = 0` or, `pdVn + Vdp = 0`(where `(k - 1)/(k) = n = ratio`) Diving both the sides by `pV` `n (dV)/(V)+(dp)/(p) = 0` On integrating `n 1n V + 1 n p = 1n C` (where `C` is constant) or, `1 n(pV^n) = 1n C` or, `pV^n = C (const)`. |
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