The molar heat capacity C for an ideal gas going through a processs which is given by C=aT, where ‘a’ is a constant and T is absolute temperature. If γ=CPCV, the work done by one mole of gas during heating from temperature T0 to ηT0 through the given process will be

ΔQ=∫T0ηT0 CdT=∫T0ηT0 aTdT=a[lnηT0-lnT0] =a lnηT0T0 = alnηΔU=CvΔT=Rγ−1(η−1)T0∴ΔW=ΔQ−ΔU=aln⁡η−η−1γ−1RT0

The molar heat capacity C for an ideal gas going through a processs which is given by C=aT, where ‘a’ is a constant and T is absolute temperature. If γ=CPCV, the work done by one mole of gas during heating from temperature T0 to ηT0 through the given process will be

ΔQ=∫T0ηT0 CdT=∫T0ηT0 aTdT=a[lnηT0-lnT0] =a lnηT0T0 = alnηΔU=CvΔT=Rγ−1(η−1)T0∴ΔW=ΔQ−ΔU=aln⁡η−η−1γ−1RT0

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The molar heat capacity C for an ideal gas going through a processs which is given by $C = \frac{a}{T}$ , where ‘a’ is a constant and T is absolute temperature. If $γ = \frac{C_{P}}{C_{V}}$ , the work done by one mole of gas during heating from temperature $T_{0}$ to $η T_{0}$ through the given process will be

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$\frac{1}{a} \ln η$

$a \ln η - (\frac{η - 1}{γ - 1}) R T_{0}$

$a \ln η - (γ - 1) R T_{0}$

$a \ln η$

answer is B.

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Detailed Solution

$\begin{array}{l} ΔQ = \int_{T_{0}}^{η T_{0}} CdT = \int_{T_{0}}^{η T_{0}} \frac{a}{T} dT = a [\ln ({ηT}_{0}) - \ln (T_{0})] = a \ln (\frac{η T_{0}}{T_{0}}) = al n (η) \\ ΔU = C_{v} ΔT = \frac{R}{γ - 1} (η - 1) T_{0} \\ ∴ ΔW = ΔQ - ΔU = al n (η) - [\frac{η - 1}{γ - 1}] {RT}_{0} \end{array}$