Important Points to Remember}

(i) \(C_{p}-C_{v}=\mathrm{R}=2 \mathrm{cal}=8.314 \mathrm{~J}\)

(ii) \(\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}=\frac{3}{2} \mathrm{R}\)

(iii) For monoatomic gas, \(C_{v}=3\) calories

(iv) For monoatomic gases, \(\frac{C_{p}}{C_{v}}=\frac{3}{5}=1.66\)

(v) For diatomic gases, \(=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}=1.40\)

\(\left(\mathrm{C}_{\mathrm{v}}=\frac{5}{3} \mathrm{R}, \mathrm{C}_{\mathrm{p}}=\frac{7}{2} \mathrm{R}\right)\)

(vi) For Triatomic gases \(\frac{C_{p}}{C_{v}}=1.33\)

- Viscosity

(i) Effect of temperature on viscosity: Viscosity decreases with increase in temperature. The relationship between coefficient of viscosity, \(\eta\) and absolute temperature \(\mathrm{T}\) is \(\eta=A \mathrm{e}^{\mathrm{Ea} / \mathrm{RT}}\) where \(A\) and \(E_{a}\) are constants for a given liquid

(ii) Effect of pressure on viscosity: Viscosity increases with increase in pressure.

- Collision frequency : The total number of collisions occurring in a unit volume of a gas per second under given set of conditions.

\(\mathrm{z} \propto \mathrm{T}^{2 / 3}\) (at constant

\(\mathrm{Z} \propto \mathrm{P}^{2}\) (at constant \(\mathrm{T}\) )

- Joule-Thomson effect : When a compressed gas is allowed to expand through a small orifice, cooling effect is caused and temperature falls. This is known as Joule Thomson effect which is observed only if the gas below certain temperature called inversion temperature.

- Greater the value of ' \(a\) ' more easily the gas is liquefiable \(a=\frac{{ }^{2}}{n^{2}}\)

\[

a=\frac{P V^{2}}{n^{2}}

\]

- Density and molar mass of a gas: According to the ideal gas equation \(P V=n R T\),

\[

\frac{\mathrm{P}_{1} \mathrm{~V}_{1}}{\mathrm{~T}_{1}}=\frac{\mathrm{P}_{2} \mathrm{~V}_{2}}{\mathrm{~T}_{2}}

\]

In terms of density, we get

\[

\mathrm{P}=\frac{\mathrm{m}}{\mathrm{M}_{\mathrm{w}}} \frac{\mathrm{RT}}{\mathrm{V}}=\frac{\mathrm{m}}{\mathrm{V}} \frac{\mathrm{RT}}{\mathrm{M}_{\mathrm{w}}}=\mathrm{d} \frac{\mathrm{RT}}{\mathrm{M}_{\mathrm{w}}}

\]

- STP (Standard temperature and pressure) or NTP (normal temperature and pressure) conditions are \(\mathrm{T}=\) \(0^{\circ} \mathrm{C}\)

\(=273.15 \mathrm{~K}, \mathrm{P}=1 \mathrm{~atm}\) or \(\mathrm{T}=0{ }^{\circ} \mathrm{C}=273.15 \mathrm{~K}, \mathrm{P}=1 \mathrm{bar}\)

(ii) Volume of a gas at STP \((\mathrm{P}=1 \mathrm{bar})=22.71098 \mathrm{~L} \mathrm{~mol}_{-1}=22.7 \mathrm{~L} \mathrm{~mol}_{-1}\)

\(1 \mathrm{~atm}=76 \mathrm{~cm}=760 \mathrm{~mm}=760\) torr

\(=101325 \mathrm{~Pa}\) or \(\mathrm{N} \mathrm{m}_{-2}=1.01325^{-} 1_{-5} \mathrm{~Pa}\) or \(\mathrm{N} \mathrm{m}_{-2}=10_{5} \mathrm{~Pa}\) or \(\mathrm{N} \mathrm{m}-2\)

\(1 \mathrm{~atm}=1.013 \mathrm{bar}\)

\(1 \mathrm{bar}=0.987 \mathrm{~atm}=10_{2} \mathrm{~K} \mathrm{~Pa}\)

\(1 \mathrm{~atm}=0.06805 \mathrm{psi}\)

\(1 \mathrm{Nm}_{-2}=6894.8 \mathrm{psi}\)

D According to Graham's Law of diffusion, under similar conditions of temperature and pressure, if \(r_{1}\) and \(r_{2}\) are rates of diffusion of two gases with densities \(d_{1}\) and \(d_{2}\) then

\[

\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}=\sqrt{\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}}=\sqrt{\frac{\mathrm{M}_{\mathrm{w} 2}}{\mathrm{M}_{\mathrm{w} 1}}}

\]

For gases at different pressures, \(\left(\mathrm{r} \propto \frac{\mathrm{P}}{\sqrt{\mathrm{M}_{\mathrm{w}}}}\right) \frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}=\frac{\mathrm{P}_{1}}{\mathrm{P}_{2}} \sqrt{\frac{\mathrm{M}_{\mathrm{w}_{1}}}{\mathrm{M}_{\mathrm{w}_{2}}}}\)

\(\triangleright\) For gases at different temperatures, \(\left(\mathrm{r} \propto \sqrt{\frac{\mathrm{T}}{\mathrm{M}_{\mathrm{w}}}}\right) \frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}=\sqrt{\frac{\mathrm{T}_{1} \mathrm{M}_{\mathrm{w}_{1}}}{\mathrm{~T}_{2} \mathrm{M}_{\mathrm{w}_{1}}}}\)

Most probable velocity, \(\alpha=\sqrt{\frac{2 \mathrm{RT}}{\mathrm{M}_{\mathrm{w}}}}\)

A) Average velocity, \(\mathrm{n}=\mathrm{v}=\sqrt{\frac{8 \mathrm{RT}}{\pi \mathrm{M}_{\mathrm{w}}}}\)

\(\Delta \mathrm{RMS}\) velocity, \(\mathrm{u}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}_{\mathrm{w}}}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}_{\mathrm{w}}}}=\sqrt{\frac{3 \mathrm{P}}{\mathrm{d}}}\)

- A gas can be liquefied by cooling or by applying pressure or by the combined effect of both. However, the effect of temperature is more important because for every gas there is a particular temperature above which it cannot be liquefied howsoever high pressure is applied.

Critical temperature \(T_{c}=\frac{8 a}{27 R b}\)

D Critical pressure: \(\quad \mathrm{P}_{\mathrm{c}}=\frac{\mathrm{a}}{27 \mathrm{~b}^{2}}\)

D Critical volume \(\left(V_{c}\right) \quad \mathrm{V}_{\mathrm{c}}=3 \mathrm{~b}\)

D) Critical compressibility factor: \(\quad \mathrm{Z}_{\mathrm{c}}=\frac{\mathrm{P}_{\mathrm{c}} \mathrm{V}_{\mathrm{c}}}{\mathrm{RT}_{\mathrm{c}}}=\frac{3}{8}\)