13 4 Average Translational Kinetic Energy – Class 12TH Physics

13.4 Average Translational Kinetic Energy of a Gas

We have calculated the pressure of the gas molecule with $n$ ( $=N/V$ ) molecules by using the gas law $PV$ ( $=nRT$ ) and kinetic theory.

PV = \frac{N}{3}m\langle v^{2}\rangle

However, $M/m$ is simply Avogadro's number i.e. $N_{A} = M m^{-1}$ . The gas constant $R$ is related with Boltzmann constant and Avogadro number. $k_{B} = R/N_{A}$

Equating these equations, we have

nRT = \frac{N}{3}m\langle v^{2}\rangle = \frac{N}{3}mv_{\mathrm{rms}}^{2} \quad \ldots (13.15)

mv_{\mathrm{rms}}^{2} = \frac{3nRT}{N}

where $\dfrac{N}{n} = N_{A}$ is the number of molecules in one mole and it is Avogadro number.

Boltzmann factor $mv_{\mathrm{rms}}^{2} = \dfrac{3RT}{N_{A}}$

\frac{1}{2}mv_{\mathrm{rms}}^{2} = \frac{3RT}{2N_{A}}

v_{\mathrm{rms}}^{2} = \frac{3RT}{mN_{A}}

v_{\mathrm{rms}} = \sqrt{\frac{3RT}{mN_{A}}} \quad \ldots (13.16)

We rewrite the pressure equation as

\frac{1}{2}m\langle v^{2}\rangle = \frac{3}{2}k_{B}T \quad \ldots (13.17)

In this equation, $\tfrac{1}{2}mv_{\mathrm{rms}}^{2}$ is the average translational kinetic energy and it is denoted by $\langle\mathrm{K.E.}_{T}\rangle$ . Therefore, the above equation is written as:

P = \frac{2}{3}\langle\mathrm{K.E.}_{T}\rangle

\langle\mathrm{K.E.}_{T}\rangle = \frac{1}{2}mv_{\mathrm{rms}}^{2}

Substituting $v_{\mathrm{rms}}^{2}$ value in Eq. (13.17)

\frac{2}{3}\left(\frac{1}{2}mv_{\mathrm{rms}}^{2}\right) = k_{B}T \quad \ldots (13.18)

\langle\mathrm{K.E.}_{T}\rangle = \frac{3}{2}k_{B}T

The average translational kinetic energy of the molecules in a gas is given by a simple constant times the temperature. So, if this model is accurate, the temperature of a gas is a direct measure of the average translational kinetic energy of its molecules.

For an air molecule at room temperature ( $300\,\mathrm{K}$ ), the quantity $k_{B}T$ is $(1.38 \times 10^{-23}\,\mathrm{J\,K^{-1}} \times 300\,\mathrm{K}) = 4.14 \times 10^{-21}\,\mathrm{J}$ . For such a small quantity, we use electron volt instead of joule (J). The electron volt is the kinetic energy of an electron that has been accelerated through a voltage difference of one volt: $1\,\mathrm{eV} = 1.6 \times 10^{-19}\,\mathrm{J}$ .

Boltzmann constant is $8.62 \times 10^{-5}\,\mathrm{eV\,K^{-1}}$ , so at room temperature, value of $k_{B}T = (8.62 \times 10^{-5}\,\mathrm{eV\,K^{-1}})(300\,\mathrm{K}) = 0.026\,\mathrm{eV}$ .

For your information

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Kinetic theory helps to understand the interaction between pollutants and the atmosphere. This knowledge is used to develop the systems to monitor the quality of air.

Daily Life Examples of Kinetic Theory of Gases

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Examples from daily life are the popcorn and pressure in tyre of a car. When popcorn kernel is heated, the moisture trapped in popcorn kernel becomes steam. The pressure in the kernel is increased and it ruptures the corn cover releasing the gelatinous starch. It becomes solid after cooling.

When air is pumped into the tyres of a car, the number of air molecules increases which raises the air pressure inside the tyres. We know that more molecules result in frequent molecular collisions with tyre's walls due to limited volume. The pressure gives the tyre its hardness and bears the car's weight without deflating.

(Fig. 13.3)

13.4 Average Translational Kinetic Energy of a Gas

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