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24/01/2022 1:42 pm
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For an ideal gas the instantaneous change in pressure 'p' with volume 'v' is given by the equation dp/dv = -ap. If p = p0 at v = 0 is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here R is the gas constant)
(1) \(\frac{p_0}{aeR}\)
(2) \(\frac{ap_0}{eR}\)
(3) Infinity
(4) 0°C
1 Answer
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24/01/2022 1:52 pm
Correct answer: (1) \(\frac{p_0}{aeR}\)
Explanation:
\(\int_{p_0}^p \frac{dp}{P}\) = \(-a \int_0^v dv\)
ℓn\(\Big(\frac{p}{p_0}\Big)\) = -av
p = p0e-av
For temperature maximum p-v product should be maximum
T = \(\frac{pv}{nR}\) = \(\frac{p_0 ve^{-av}}{R}\)
\(\frac{dT}{dv}\) = 0 ⇒ \(\frac{p_0}{R}\){e-av + ve-av(-a)}
\(\frac{p_0e^{-av}}{R}\){1 - av} = 0
v = \(\frac{1}{a}\), ∞
T = \(\frac{p_01}{Rae}\) = \(\frac{p_0}{Rae}\)
at v = ∞
T = 0