Consider one mole of an ideal gas whose volume changes with temperature as V=αT , where α is constant. Find the amount of heat required to raise its temperature by ΔT , if its adiabatic constant is γ.

Given that TV=constantcompare with TVx−1=constantwe get x−1=1 x=2now C=CV+R1−x=CV+R1−2⇒C=CV−RSince Q=nCΔT=(1)CΔTQ=(CV−R)ΔT =CVΔT−RΔT =Rγ−1ΔT−RΔT=RΔT[1γ−1−1]∴Q=(γ-2)RΔT1−γ

Consider one mole of an ideal gas whose volume changes with temperature as V=αT , where α is constant. Find the amount of heat required to raise its temperature by ΔT , if its adiabatic constant is γ.

Given that TV=constantcompare with TVx−1=constantwe get x−1=1 x=2now C=CV+R1−x=CV+R1−2⇒C=CV−RSince Q=nCΔT=(1)CΔTQ=(CV−R)ΔT =CVΔT−RΔT =Rγ−1ΔT−RΔT=RΔT[1γ−1−1]∴Q=(γ-2)RΔT1−γ

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Consider one mole of an ideal gas whose volume changes with temperature as $V = \frac{α}{T}$ , where $α$ is constant. Find the amount of heat required to raise its temperature by $Δ T$ , if its adiabatic constant is $γ$ .

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$\frac{1}{1 - γ}$

$\frac{(γ - 2) R Δ T}{1 - γ}$

$\frac{γ}{1 - γ}$

$\frac{γ Δ T}{1 - γ}$

answer is B.

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Detailed Solution

Given that TV=constant

$\begin{array}{l} c o m p a r e w i t h T V^{x - 1} = c o n s t a n t \\ w e g e t x - 1 = 1 \\ x = 2 \\ n o w C = C_{V} + \frac{R}{1 - x} = C_{V} + \frac{R}{1 - 2} \\ \Rightarrow C = C_{V} - R \\ S i n c e Q = n C Δ T = (1) C Δ T \\ Q = (C_{V} - R) Δ T \\ = C_{V} Δ T - R Δ T \\ = \frac{R}{γ - 1} Δ T - R Δ T = R Δ T [\frac{1}{γ - 1} - 1] \\ ∴ Q = \frac{(γ - 2) R Δ T}{1 - γ} \end{array}$