7.20: Adiabatic Expansions of An Ideal Gas

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Consider an ideal gas that undergoes a reversible adiabatic expansion from an initial state, specified by known values \(V_1\) and \(T_1\), to a new state in which the value of the volume, \(V_2\), is known but the value of the temperature, \(T_2\), is not known. For an adiabatic reversible process, \(q=0\), and \(w=\Delta E\). Since \({\left({\partial E}/{\partial T}\right)}_V=C_V\), we have \(dE=C_VdT\), so that

\[w=\Delta E=\int^{T_2}_{T_1}{C_V}dT \nonumber \]

For any gas, we can assume that \(C_V\) is approximately constant over a small temperature range. Taking \(C_V\) to be constant in the interval \(T_1<t_2\)>, we have \(w=\Delta E=C_V\left(T_2-T_1\right)\). We obtain the enthalpy change from

\[\Delta H=\Delta E+\Delta \left(PV\right)=\Delta E+\Delta \left(RT\right)=C_V\left(T_2-T_1\right)+R\left(T_2-T_1\right)=C_P\left(T_2-T_1\right) \nonumber \]

where we use our ideal-gas result from Section 7.16, \(C_P=C_V+R\).

While these relationships yield the values of the various thermodynamic quantities in terms of the temperature difference, \(T_2-T_1\), we have yet to find the final temperature, \(T_2\). To find \(T_2\), we return to the first law: \(dE=dq+dw\). Substituting for \(dE\), \(dq\), and \(dw\), and making use of the ideal gas equation, we have

\[C_VdT=-PdV=-\frac{RT}{V}dV \nonumber \]

from which, by separation of variables, we have

\(\int^{T_2}_{T_1}{C_V}\frac{dT}{T}=-R\int^{V_2}_{V_1}{\frac{dV}{V}}\) (one mole ideal gas, reversible adiabatic expansion)

If we know \(C_V\) as a function of temperature, we can integrate to find a relationship among \(T_1\), \(T_2\), \(V_1\), and \(V_2\). Given any three of these quantities, we can use this relationship to find the fourth. If \(C_V\) is independent of temperature, as it is for a monatomic ideal gas, we have

\[ \ln \frac{T_2}{T_1} =-\frac{R}{C_V} \ln \frac{V_2}{V_1} =\frac{R}{C_V} \ln \frac{V_1}{V_2} = \ln \left(\frac{V_1}{V_2}\right)^{R/C_V} \nonumber \]

so that

\[\frac{T_2}{T_1}= \left(\frac{V_1}{V_2}\right)^{R/C_V} \nonumber \] (monatomic ideal gas, reversible adiabatic expansion)

For the spontaneous adiabatic expansion of an ideal gas against a constant applied pressure, we have \(dq=0\), so that \(dE=dw\), and \(C_VdT=-P_{applied}dV\). Given the initial conditions, we can find the final temperature from

\[\int^{T_2}_{T_1}{C_VdT}=\int^{V_2}_{V_1}{-P_{applied}dV}=-P_{applied}\left(\frac{RT_2}{P_{applied}}-\frac{RT_1}{P_1}\right)=R\left(\frac{P_{applied}T_1}{P_1}-T_2\right) \nonumber \] (spontaneous adiabatic process)

The changes in the remaining state functions can then be calculated from the relationships above. In this spontaneous adiabatic process, all of the other thermodynamic quantities are different from those of a reversible adiabatic process that reaches the same final volume.