The equation of state of a certain gas is (k is constant)

$v = \frac{R T}{P} + \frac{k}{R T}$

Then the value of k for the process of throttling of gas from initial pressure $P_{1}$ and temperature $T_{1}$ to final pressure $P_{2}$ and tempertaure $T_{2}$ is

(\frac{T_{2}^{2} - T_{1}^{2}}{4 (P_{1} - P_{2})}) R c_{p}

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(\frac{T_{2}^{2} - T_{1}^{2}}{(P_{1} - P_{2})}) R c_{p}

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(\frac{T_{2}^{2} - T_{1}^{2}}{2 (P_{1} - P_{2})}) R c_{p}

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((\frac{R}{c_{p}}) \frac{T_{2}^{2} - T_{1}^{2}}{4 (P_{1} - P_{2})})

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Solution

The correct option is A $(\frac{T_{2}^{2} - T_{1}^{2}}{4 (P_{1} - P_{2})}) R c_{p}$
$d h = c_{p} d T + [v - T {(\frac{δ v}{δ T})}_{p}] d p$

during throttling dh = 0 and the above equations gives

$c_{p} d T = - [v - T {(\frac{δ v}{δ T})}_{p}] d v$

Noe from the given equation of state

${(\frac{δ v}{δ T})}_{p} = \frac{R}{P} - \frac{k}{R T^{2}}$

Substituting thus in the above equation,

$c_{p} d T = - [\frac{R T}{P} + \frac{k}{R T} - \frac{k}{R T}] d p$

$= \frac{- 2 k}{R T} d p$

on rearanging

$\frac{T d t}{k} = \frac{- 2 c P}{R c_{p}}$

$\frac{T_{2}^{2} - T_{1}^{2}}{2 k} = \frac{- 2 (P_{2} - P_{1})}{R C_{p}}$

$\frac{T_{2}^{2} - T_{1}^{2}}{4 k} = \frac{P_{1} - P_{2})}{R C_{p}}$

$k = (\frac{T_{2}^{2} - T_{1}^{2}}{4 (P_{1} - P_{2})}) R c_{p}$

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The equation of state of a certain gas is (k is constant) v=RTP+kRT Then the value of k for the process of throttling of gas from initial pressure P1 and temperature T1 to final pressure P2 and tempertaure T2 is

The equation of state of a certain gas is (k is constant)

$v = \frac{R T}{P} + \frac{k}{R T}$

Then the value of k for the process of throttling of gas from initial pressure $P_{1}$ and temperature $T_{1}$ to final pressure $P_{2}$ and tempertaure $T_{2}$ is