[[Thermodynamics]] Modifications to the [[Ideal Gas Law]] ## Estimating V with [[Compressibility Factor|Z]]: Equations of State #### Differential of V(T,P): $\begin{gather}\frac{dV}{V} = BdT - KdP \end{gather}$ B = coeff of thermal expansivity K = coeff of isothermal expansion For constant P or T → integrate either BdT or -KdP $\begin{gather}ln(\frac{V_2}{V_1}) = B(T_2 - T_1) \\ or -K(P_2 - P_1)\end{gather}$ ==Use==: Given constants B and K, calculate volume change for changes in pressure and temperature under close to ideal conditions #### Rackett Equation: V_sat = V_cZ_c^{(1-T_r)^.2857} ==Use==: Estimate the volume of a saturated liquid with only the temperature and information from tables (can be used for many substances) #### Virial Equation: Power series expansion of Z(T,P) in powers of 1/V $\begin{gather}Z = 1 + \frac{B}{V}+ \frac{C}{V^2}...\end{gather}$ #### Truncated Virial Equation: $\begin{gather}Z = 1 + \frac{BP}{RT}\end{gather}$ ==Use==: Linear approximation based on virial equation. Use on gases with low P_R and Z > .75, or when values of P_R and T_R ([[Reduced Temperature and Pressure]]) stay in the linear part of the ZP graph. ##### Pitzer Correlation: Estimate B using T $\begin{gather}B_0 = .083 - \frac{.422}{T_R^{1.6}} \\ B_1 = .139 - \frac{.172}{T_R^{4.2}} \\ B = \frac{R T_c}{P_c}{B_0 + ωB_1}\end{gather}$ Note: B is a function of T only, Z is a function of P and T #### Lee-Kesler Tables: Tabulated version of ZP graphs Calculate P_R and T_R , use tables to interpolate z₀ and z₁ Z = z₀ + ωz₁ ==Use: Less math-intensive way to estimate compressability #### Van der Waals Equation: $\begin{gather}P = \frac{RT}{V-b} - \frac{a}{V^2} \end{gather}$ Note: a and b depend on substance properties, not T or P $\begin{gather}a = \frac{27(RT_c)^2}{P_c} \qquad b = \frac{RT_c}{8P_c} \end{gather}$ ==Use==: Model of attractive and repulsive forces, first good model but not used for accurate calculations because it doesn’t take P or T into account #### SRK Equation: $\begin{gather}P = \frac{RT}{V-b} - \frac{a}{V(V+b)} \end{gather}$ $\begin{gather}𝛀 = .480 + 1.574ω - .176ω^2 \\ a = .4278 \frac{{R^2}{T_c^2}}{Pc}[1 +𝛀(1-T_R^{1/2})]^{2} \\ b = .0866 \frac{RT_c}{P_c}\end{gather}$ $\begin{gather}Set \quad V = \frac{ZRT}{P} \quad → \quad Z^3 - Z^2 + (A-B-B^2)Z - AB = 0\end{gather}$ $\begin{gather}A = \frac{aP}{(RT)^2} \qquad B = \frac{bP}{RT}\end{gather}$ Solving the cubic equation for Z will give 3 roots. The lowest root corresponds to saturated liquid, the highest (should be close to 1) to saturated vapor. ==Use==: A more accurate estimate of volume under non-ideal conditions #### PR Equation: $\begin{gather}P = \frac{RT}{V-b} - \frac{a}{V^2 + 2bV + b^2} \end{gather}$ Like SRK, a is a function of T_c, P_c, ω, T_R and b is a function of T_c and P_c. Note: Coefficients $a$ and $b$ are different in VDW, SRK, and PR equations, but $a$ always corresponds to attraction and b corresponds to repulsion.