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# Lecture 3: Equation of state ## 3.1 Introduction ### 3.1.1 Definition The topic of today's lecture is the equation of state which relates state variables to each other. $$ f(P, V, T, N) = 0 $$ In general, it is very difficult to find the exact equation of state analytically. ### 3.1.2 Exampl...

# Lecture 3: Equation of state ## 3.1 Introduction ### 3.1.1 Definition The topic of today's lecture is the equation of state which relates state variables to each other. $$ f(P, V, T, N) = 0 $$ In general, it is very difficult to find the exact equation of state analytically. ### 3.1.2 Example The best-known example of an equation of state is the ideal gas law $$ PV = Nk_BT $$ where * $P$ is the pressure * $V$ is the volume * $N$ is the number of particles * $T$ is the temperature * $k_B$ is the Boltzman constant ## 3.2 Van der Waals Equation ### 3.2.1 Motivation The van der Waals equation of state is a more realistic equation of state than the ideal gas law. The ideal gas law assumes that the particles are point particles and that there are no interactions between the particles. The van der Waals equation of state takes into account the finite size of the particles and the attractive interactions between the particles. ### 3.2.2 Derivation $$ (P + a\frac{N^2}{V^2})(V - Nb) = Nk_BT $$ where * $a$ is a measure of the attractive interactions between the particles * $b$ is a measure of the finite size of the particles #### Excluded Volume The excluded volume is the volume that is not available to the particles because of the finite size of the particles. The excluded volume is proportional to the number of particles. $$ V_{excluded} = Nb $$ #### Mean-field theory of interaction The attractive interactions between the particles reduce the pressure. The reduction in pressure is proportional to the square of the density. $$ P_{reduction} = a\frac{N^2}{V^2} $$ ### 3.2.3 isotherms Isotherms are curves of constant temperature in a $P-V$ diagram. #### Critical point The critical point is the point at which the isotherm has a horizontal tangent. At the critical point, the first and second derivatives of the pressure with respect to the volume are zero. $$ \frac{\partial P}{\partial V} = 0 $$ $$ \frac{\partial^2 P}{\partial V^2} = 0 $$ $$ \begin{cases} P_c = \frac{a}{27b^2} \\ V_c = 3Nb \\ T_c = \frac{8a}{27k_Bb} \end{cases} $$ ### 3.2.4 Reduced units Reduced units are units that are scaled by the critical point values. $$ \tilde{P} = \frac{P}{P_c} $$ $$ \tilde{V} = \frac{V}{V_c} $$ $$ \tilde{T} = \frac{T}{T_c} $$ Then the vdW equation becomes $$ (\tilde{P} + \frac{3}{\tilde{V}^2})(3\tilde{V} - 1) = 8\tilde{T} $$ This equation is universal, i.e. it does not depend on the specific gas. ### 3.2.5 Maxwell construction The Maxwell construction is a method for finding the equilibrium pressure in a coexistence region. The Maxwell construction is based on the fact that the Gibbs free energy is the same in both phases in a coexistence region. The area above and below the equilibrium pressure line are equal. ## 3.3 Virial Expansion ### 3.3.1 Motivation The virial expansion is a power series expansion of the equation of state in terms of the density. The virial expansion is useful for describing real gases at low densities. ### 3.3.2 Expansion $$ \frac{P}{k_BT} = \rho + B_2(T)\rho^2 + B_3(T)\rho^3 +... $$ where * $\rho = N/V$ is the density * $B_i(T)$ are the virial coefficients ### 3.3.3 Example For hard spheres, the second virial coefficient is $$ B_2(T) = \frac{2\pi}{3}d^3 $$ where $d$ is the diameter of the hard spheres.