Kinetic interpretation of temperature

Sacademy June 19, 2023 1 min read

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Kinetic interpretation of temperature

Pressure of an ideal gas

$P=\frac{1}{3}\rho \bar{c^{2}}$

$or\;\;\;\;\;P=\frac{1}{3}\frac{M}{V} \bar{c^{2}}\;\;\;\;\;[\because \rho =M/V]$

$or\;\;\;\;\;PV=\frac{1}{3}M \bar{c^{2}}$

∵ M= mN

m = mass of one molecule
N = 1 number of molecule in 1 gm-molecule (Avogadro number)

$\therefore \;\;\;\;\;PV=\frac{1}{3}mN \bar{c^{2}}$

∵ For 1 gm-molecule of a gas at temperature T, PV = RT

V = Volume of gas molecule of gas
R = Universal gas constant

$\therefore \;\;\;\;\;\frac{1}{3}mN \bar{c^{2}}=RT$

$or \;\;\;\;\;\frac{1}{2}\times \frac{2}{3}mN \bar{c^{2}}=RT$

$or \;\;\;\;\;\frac{1}{2} m \bar{c^{2}}=\frac{3}{2}\frac{R}{N}T$

$or \;\;\;\;\;\frac{1}{2} m \bar{c^{2}}=\frac{3}{2}kT$

where k is Boltzmann’s constant

$Now \;\;\;\;\; \bar{c^{2}}=\frac{3kT}{m}$

$or \;\;\;\;\; c_{rms}=\sqrt{\bar{c^{2}}}=\sqrt{\frac{3kT}{m}}$

$or \;\;\;\;\; c_{rms}=\sqrt{T}$

$(1/2) m \bar{c^{2}}=\bar{E}$ is Kinetic energy of a molecule of a gas

$\because \;\;\;\;\;\frac{1}{2} m \bar{c^{2}}=\frac{3}{2}kT$

$\therefore \;\;\;\;\; \bar{E}\propto T\;\;\;and\;\;\;\bar{c^{2}}\propto T$

Conclusion

The mean kinetic energy of a molecule is directly proportional to the absolute temperature of the gas.
The mean square velocity of the molecule is directly proportional to the absolute temperature of the gas.
The amount of heat produced by a body is equal to the total kinetic energy of its molecules and its temperature is proportional to the mean kinetic energy of its molecules.
This is the kinetic interpretation of temperature.

Perfect gas equation

$\because \;\;\;\;\;P=\frac{1}{3}\rho \bar{c^{2}}=\frac{1}{3}\frac{mn}{V}\bar{c^{2}}$

$\because \;\;\;\;\;PV=\frac{1}{3}mn\bar{c^{2}}$

$or \;\;\;\;\;PV=\frac{2}{3} \times \frac{1}{2}mn\bar{c^{2}}\;\;\;\;\;...(1)$

$But \;\;\;\;\;\frac{1}{2}mn\bar{c^{2}}\propto T$

$\therefore \;\;\;\;\;\frac{1}{2}mn\bar{c^{2}}=\alpha T\;\;\;\;\;...(2)$

From equations (1) and (2), we get

$PV=\frac{2}{3} \alpha T$

But 2α/3 = R (Universal gas constant)

$\therefore\;\;\;\;\;PV=RT$

This is known as perfect gas equation or ideal gas equation

To get the detailed knowledge about the kinetic interpretation of temperature click here.

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