# Formulas of Gaseous State

Boyle’s law,

$P \propto \dfrac{1}{V} PV = Constant$

$P_1V_1 = P_2V_2$  (At constant temperature)

Charle’s law;

$V \propto T,\\[3mm] \dfrac{V}{T} = \text{Constant} \\ \dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}\text{At constant pressure}$

Gas equation,

$\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$

Ideal gas equation, PV = nRT.

$V \propto N$ (At const. temperature and pressure)

Dalton’s Law,

$P = p_1 + p_2 + p_3 \cdots$

Graham’s law,

$\dfrac{r_1}{r_2} = \sqrt{\dfrac{d_2}{d_1}} = \sqrt{\dfrac{M_2}{M_1}}$

Kinetic gas equation;

$PV = \dfrac{1}{3}mnu^2$

Root mean square velocity;

$u = \sqrt{\dfrac{3PV}{M}} = \sqrt{\dfrac{3RT}{M}} = \sqrt{\dfrac{3P}{d}}$

Average velocity;

$v = \sqrt{\dfrac{8RT}{\pi M}} = \sqrt{\dfrac{8PV}{\pi M}} = \sqrt{\dfrac{8P}{ \pi d}}$

Most probable velocity

$\alpha = \sqrt{\dfrac{2RT}{M}} = \sqrt{\dfrac{2PV}{M}} = \sqrt{\dfrac{2P}{d}}$

U : v : $\alpha$ = 1.0 : 0.9213 : 0.8177

$\alpha$ : v : u = 1.0 : 1.128 : 1.234

Van der Waals equation,

$[P + \dfrac{an^2}{V^2}] [V- nb] = nRT$

Critical components,

$V_c = 3b \\[3mm] P_c = \dfrac{a}{27 b^2} \\[3mm] T_c = \dfrac{8a}{27 bR} \\[3mm] \dfrac{P_cV_c}{T_c} = \dfrac{3}{8}R \\[3mm] a = 3V_c^2P_c = \dfrac{27 R^2T_c^2}{64 P_c} \\[3mm] b = \dfrac{V_c}{3} = \dfrac{RT_c}{8P_c} \\[3mm] T_i = \dfrac{2a}{bR} \\[3mm] T_B = \dfrac{a}{bR}$

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