Question
The following terms are used in thermodynamics:Isothermal compressibility $\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)$Thermal expansion coefficient$$\beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)$$Show that for the ideal gas equation, $P V=R T$, we have$\kappa=\frac{1}{P}$ and $\beta=\frac{1}{T}$
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Here, \( P \) is the pressure, \( V \) is the volume, \( R \) is the ideal 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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Generally, volume expansivity $\beta$ and isothermal compressibility $\kappa$ depend on $\mathrm{T}$ and $\mathrm{P}$. Prove that: $$\left(\frac{\partial \beta}{\partial P}\right)_{T}=-\left(\frac{\partial \kappa}{\partial T}\right)_{P}$$
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