Question

show that the average kinetic energy of a gas molecule is directly proportional to the Absolute Temperature of the gas hence give the Kinetic interpretation of temperature​

Answer 1

Given that,

The average kinetic energy of a gas molecules is directly proportional to the absolute temperature of the gas.

We need to prove the given statement

Using the pressure expression

$P=\dfrac{mnv^2}{3}$

Where, m = mass of molecule

n = number of molecules per unit  volume

v = rms speed

Put the value of n

$P=\dfrac{mNv^2}{3V}$

Where, N = number of molecules

V = volume

$PV=\dfrac{mNv^2}{3}$ ...(I)

We know that,

The kinetic energy is

$E=\dfrac{1}{2}mv^2$

$mv^2=2E$

Put the value inthe equation

$PV=\dfrac{2NE}{3}$

Put the value of PV from ideal gas

$\mu RT=\dfrac{2NE}{3}$

Where, $\mu=\dfrac{N}{N_{A}}$

So, $\dfrac{N}{N_{A}}RT=\dfrac{2NE}{3}$

$E=\dfrac{3}{2}kT$

Where, $\dfrac{R}{N_{A}}$ = boltzmann constant

$E\propto T$

The average kinetic energy of a gas molecules is directly proportional to the absolute temperature of the gas.

Hence, This is proved.

Answer 2

Answer:

the average kinetic energy of a gas molecule is directly proportional to the Absolute Temperature of the gas