4.0 Kinetic Interpretation of Temperature and Pressure

According to Kinetic Theory of Gases, $$\begin{equation} \begin{aligned} PV = \frac{1}{3}mNC_{rms}^2 \\ \\\end{aligned} \end{equation} $$

or, $$\begin{equation} \begin{aligned} N RT = \frac{1}{3} \times 2 \times \left( {\frac{1}{2}mC_{rms}^2} \right)N \\ \\\end{aligned} \end{equation} $$

or, $$\begin{equation} \begin{aligned} \frac{N}{{{N_A}}}RT = \frac{2}{3}N\left( {\frac{1}{2}mC_{rms}^2} \right) \Rightarrow \frac{1}{2}mC_{rms}^2 = \frac{3}{2}kT \\ \\\end{aligned} \end{equation} $$

$\because \frac{R}{{{N_A}}} = k$, Average translational kinetic energy of a gas molecule is $$\frac{3}{2}kT$$

Hence, the average translational kinetic energy of a gas molecule depend only on its temperature and is independent of its nature i.e., gas molecule whether, they are monoatomic, diatomic or polyatomic at a given temperature, have same translational kinetic energy though their $rms$ speeds are different.

$$\overline {K{E_t}} = \frac{1}{2}mC_{rms}^2 = \frac{3}{2}kT$$

i.e. $$C_{rms}^2 \propto T$$

Thus, ${\overline {K{E_t}}}$ for $N$ molecules =$ \frac{3}{2}Nkt $

$$\begin{equation} \begin{aligned} \overline {K{E_t}} {\text{ }}For{\text{ }}N{\text{ }}molecules = \frac{3}{2}N {N_A}kt \\ \\\end{aligned} \end{equation} \ \ \ \ \left[ {\because N \ mole = N {N_A}\ molecules} \right]$$$$ \Rightarrow Average\ Translational\ K.E.\ for\ N\ moles = \frac{3}{2}NRT $$$$ \Rightarrow \ Average\ Translational\ K.E.\ for\ 1\ mole = \frac{3}{2}RT $$$$ \Rightarrow {U_t} = Average\ translational\ K.E.\ for. N\ mole = \frac{3}{2}NRT$$$$ \Rightarrow E = \frac{{{U_t}}}{V} = K.E.\ per\ unit\ volume = \frac{3}{2}\frac{{N RT}}{V} = \frac{3}{2}P $$$$ \Rightarrow P = \frac{2}{3}E $$

i.e., pressure of a gas is numerically equal to $\left( {\frac{2}{3}} \right)$ times of its average translational kinetic energy per unit volume.

Now, according to kinetic theory of gases

$$\begin{equation} \begin{aligned} PV = \frac{1}{3}mNC_{rms}^2{\text{ }}\ \ and\ \ {\text{ }}C_{rms}^2 \propto T \\ \\\end{aligned} \end{equation} $$

Thus, pressure of a gas is governed by three independent factors volume($V$), temperature($T$) and mass of gas($mN$).