Question
The isothermal compressibility $\kappa$ is defined as\[\kappa=-\frac{1}{\bar{V}}\left(\frac{\partial \bar{V}}{\partial P}\right)_{T}\]Show that\[\kappa=\frac{1}{P}\]for an ideal gas.
Step 1
Step 1: We start with the ideal gas law, which is given by \[PV = nRT\] where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature. Show more…
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The cubic expansion coefficient $\alpha$ is defined by \[\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\] and the isothermal compressibility $\kappa$ is defined by \[\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\] Calculate these quantities for an ideal gas.
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The equation of state of an ideal gas is $P V=n R T$, where $n$ and $R$ are constants. (a) Show that the volume expansivity $\beta$ is equal to $1 / T$. (b) Show that the isothermal compressibility $\kappa$ is equal to $1 / P$.
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