For the case of an ideal gas, find the equation of the process in which the molar heat capacity varies as C=Cv+αT2, where α is constant. Take V0 to be the initial volume of the gas.
A
V=V0eαT22R
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B
V=V0e2αTR
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C
V=V0eαT2R
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D
V=V0eT2αR
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Solution
The correct option is AV=V0eαT22R C=Cv+αT2……(i) Also, we know that C=Cv+Pn(dVdT)……(ii) Comparing (i) and (ii) αT2=Pn(dVdT) ⇒αT/2=/nR/TV/n(dVdT){∵PV=nRT} ⇒αT=RV(dVdT) Integrating on both sides, ⇒∫dVV=∫αRTdT ⇒lnV=(αR)T22+constant For T=0,V=V0 ⇒ln(VV0)=(αR)T22 ⇒V=V0eαT22R