Physics > Kinetic Theory of Gases > 12.0 Atomicity of a gas
Kinetic Theory of Gases
1.0 Assumptions of Kinetic Theory of Gases
2.0 Ideal Gas
3.0 Expression For Pressure
4.0 Kinetic Interpretation of Temperature and Pressure
5.0 Ideal Gas Laws
6.0 Maxwells's Distribution
7.0 Gas Speed
8.0 Degrees of Freedom
9.0 Law of Equipartition of Energy
10.0 Heat Capacity
11.0 Relation between Degree of Freedom and Specific Heat of Gas
12.0 Atomicity of a gas
12.1 Molar heat capacity of different gases
If ${C_P}$ and ${C_V}$ are molar heat capacities at constant pressure and constant volume, then for ideal gas $${C_P} - {C_V} = R$$$$\gamma = \frac{{{C_P}}}{{{C_V}}} $$ and Internal energy $(U)$ is $$ U=N {C_V}T$$
Also, if $f$ degrees of freedom for a gas, then its internal energy is $$ U= \frac{f}{2}NRT$$
Thus,
$$N{C_V}T = \frac{f}{2}NRT$$
$$ \Rightarrow {{C_V} = \frac{f}{2}R}$$
$$\because {C_P} - {C_V} = R$$
$$\Rightarrow {C_P} = {C_V} + R \Rightarrow {C_P} = (f + 2)\frac{R}{2} $$
$$ \because \gamma = \frac{{{C_P}}}{{{C_V}}} = \frac{{f + 2}}{f} \Rightarrow \gamma = 1 + \frac{2}{f} $$
Question 4. If 3 moles $H_2$ gas is mixed with 2 moles of helium gas, then calculate equivalent molar mass of the as and adiabatic exponent of the mixture.
Solution: Given, ${N_1}=3$, ${M_01}=2\ gm$, ${N_2}=2$, ${M_02}=4\ gm$, ${f_1}=5$, ${f_2}=3$
Equivalent molar mass is $$ = \frac{{{N_1}{M_{01}} + {N_2}{M_{02}}}}{{{N_1} + {N_2}}} = \frac{{(3 \times 2) + (2 \times 4)}}{{3 + 2}} = \frac{{6 + 8}}{5} = \frac{{14}}{5}\ gm$$
$$\begin{equation} \begin{aligned} \therefore {M_0} = \frac{{14}}{5} = 2.8\;gm/mole \\ \\\end{aligned} \end{equation} $$
Let degree of freedom of mixture is $f$, then
$$\begin{equation} \begin{aligned} f = \frac{{{\mu _1}{f_1} + {\mu _2}{f_2}}}{{{\mu _1} + {\mu _2}}} = \frac{{(3 \times 5) + (2 \times 3)}}{{3 + 2}} = \frac{{15 + 6}}{5} = \frac{{21}}{5} \\ \\\end{aligned} \end{equation} $$
Thus $$\begin{equation} \begin{aligned} \gamma = 1 + \frac{2}{f} = 1 + \frac{{2 \times 5}}{{21}} = \frac{{32}}{{21}} = 1.476 \\ \\\end{aligned} \end{equation} $$
Question 5. When a certain amount of heat is supplied to a diatomic gas at constant volume, rise in temperature is $T_0$. When same heat is supplied to the gas at constant pressure, find the rise in temperature.
Solution: Let the supplied be $Q$. In first case $Q = n{C_v}({T_0})$ and in second case $Q = n{C_p}({T_0'})$
$$ \Rightarrow T_0' = {T_0} \times \frac{{{C_v}}}{{{C_p}}} = \frac{{5{T_0}}}{7}$$
