Adiabatic Pathways

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• An adiabatic pathway is one in which no heat is transfered, \(dq=0\).
• If an ideal gas expands adiabatically, it is doing work against the surrounds (\(w<0\)) so the internal energy must drop (\(\Delta U<0\))
• Because the internal energy of an ideal gas depends solely on the temperature, if \(\Delta U<0\) then \(\Delta T<0\)

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Reversible Adiabatic Expansion of an Ideal Gas

We know that \[dw=-PdV\] and \[dw=nC_VdT\] therefore \[-PdV=nC_VdT\] Using the ideal gas law we find that \[- \frac{nRT}{V}dV=nC_VdT\]

Integrating to get to macroscopic changes \[\int_{V_1}^{V_2} \frac{dV}{V} = - \frac{C_V}{R} \int_{T_1}^{T_2} \frac{dT}{T}\] which solves to \[ln \left(\frac{V_2}{V_1} \right) =- \frac{C_V}{R} ln \left(\frac{T_2}{T_2} \right)\] \[\left(\frac{V_2}{V_1} \right) = \left(\frac{T_2}{T_1} \right) ^{\frac{C_V}{R}} \Rightarrow T_1 \left(\frac{V_2}{V_1} \right) ^{\frac{R}{C_V}} = T_2\]

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:::: {.whitebox data-latex-““} Example 3.7

\(1.00mol\) of an ideal gas (\(C_V=\frac{3}{2} R\)) initially occupies \(22.4L\) at \(273K\). The gas expands adiabatically and reversibly to a final volume of \(44.8L\). Calculate \(\Delta T\), \(q\), \(w\), \(\Delta U\), and \(\Delta H\) for the expansion. ::::