Temperature Scale:

$ \frac{C-O}{100-0}=\frac{K-273}{373-273}=\frac{F-32}{212-32}$ = $\frac{R-R(O)}{R(100)-R(O)} $

where R= Temp. on unknown scale.

Boyle’s law and measurement of pressure:

At constant temperature, $\mathrm{V}$ $\propto$ $\frac{1}{\mathrm{P}}$

$ \quad\quad \quad \quad \quad \quad \quad \quad \quad P_1 V_1 = P_2 V_2$

Charles law :

At constant pressure, $\quad V \alpha T \quad $ or $\quad \frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$

Gay-Lussac’s law:

At constant volume,

$\mathrm{P}$ $\propto$ $\mathrm{T} \quad \frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}} \rightarrow$ temp on absolute scale

Ideal gas Equation:

$$ P V=n R T $$

$$ P V=\frac{w}{m} R T \text { or } P=\frac{d}{m} R T \text { or } P m=d R T $$

Daltons law of partial pressure:

$\quad P_{1}=\frac{n_{1} R T}{v}, \quad P_{2}=\frac{n_{2} R T}{v}, \quad P_{3}=\frac{n_{3} R T}{v}$ and so on.

$\quad $ Total pressure = $ P_1 + P_2 + P_3….$

$\quad $ Partial pressure $=$ mole fraction $X$ Total pressure.

Amagat’s law of partial volume:

$$ V=V_{1}+V_{2}+V_{3}+\ldots \ldots $$

Average molecular mass of gaseous mixture:

$$ M_{\text {mix }}=\frac{\text { Total mass of mixture }}{\text { Total no. of moles in mixture }} =\frac{n_{1} M_{1}+n_{2} M_{2}+n_{3} M_{3}}{n_{1}+n_{2}+n_{3}}$$

Graham’s Law

Rate of diffusion $r \propto \frac{1}{\sqrt{d}} ; \quad d=$ density of gas

$$ \frac{r_{1}}{r_{2}}=\frac{\sqrt{d_{2}}}{\sqrt{d_{1}}}=\frac{\sqrt{M_{2}}}{\sqrt{M_{1}}}=\sqrt{\frac{V \cdot D_{2}}{V \cdot D_{1}}} $$

Kinetic Theory of Gases

$\mathrm{PV}=\frac{1}{3} \mathrm{mN} \overline{\mathrm{U}^{2}} \quad \text{Kinetic equation of gases}$

Average K.E. for one mole $=N_{A}\left(\frac{1}{2} m \overline{U^{2}}\right)=\frac{3}{2} K N_{A} T=\frac{3}{2} R T$

Root mean square speed

$U_{r m s}=\sqrt{\frac{3 R T}{M}} \quad$ molar mass must be in $\mathrm{kg} / \mathrm{mole}$

Average speed

$ U_{av} = U_{1}+U_{2}+U_{3}+ \ldots \ldots \mathrm{U}_{\mathrm{N}}$

$U_{\text {avg. }}=\sqrt{\frac{8 R T}{\pi M}}=\sqrt{\frac{8 K T}{\pi m}} \quad K$ is Boltzmann constant

Most probable speed

$U_{MPS } =\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 K T}{m}}$

Van der Waal’s equation :

$ \left(P+\frac{a n^{2}}{v^{2}}\right)(v-n b)=n R T $

Critical constants:

$\mathrm{V}_{\mathrm{c}}=3 \mathrm{~b}$,

$\mathrm{P}_{\mathrm{c}}=\frac{\mathrm{a}}{27 \mathrm{~b}^{2}}$,

$\mathrm{~T}_{\mathrm{C}}=\frac{8 \mathrm{a}}{27 \mathrm{Rb}}$