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B) For a thermodynamic system, isobaric coefficient of volume expansion (a) andisothermal compressibility (B) are defined as1 ava=)и отp1 avB=-У дрT TShow that for an isochoric change, ß dp = a dT. |
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Answer» Explanation: Given, \alpha = \frac{1}{V}(\frac{dV}{dT})_pα= V 1
( dT dV
) p
\beta = - \frac{1}{V}(\frac{dV}{DP})_Tβ=− V 1
( dP dV
) T
Now, from the ideal GAS equation, PV=nRTPV=nRT V=\frac{nRT}{P}V= P nRT
Now, taking the DIFFERENTIATION with respect to the corresponding terms, \frac{dV}{dT}=\frac{NR}{P} dT dV
= P nR
if P is constant then, (\frac{dV}{dT})_P=\frac{nR}{P}...(i)( dT dV
) P
= P nR
...(i) now, taking the differentiation with respect to P, (\frac{dV}{dP})_T=\frac{nRT}{P^2}...(ii)( dP dV
) T
= P 2
nRT
...(ii) Now, taking the ratio of (i)(i) and (ii)(ii) \frac{(\frac{dV}{dT})_P}{(\frac{dV}{dP})_T}=\frac{\frac{nR}{P}}{\frac{nRT}{P^2}} ( dP dV
) T
( dT dV
) P
= P 2
nRT
P nR
\frac{(dP)_T}{(dT)_P}=\frac{P}{T} (dT) P
(dP) T
= T P
(dP)_T=(dT)_P\times \frac{P}{T}(dP) T
=(dT) P
× T P
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