Question

Prove that the kinetic energy of a gas is directly proportional to the absolute temperature of the gas.

Answer 1

Step 1: Kinetic theory of gases

1. There is no intermolecular force between the particles.
2. The molecules are in constant random motion.
3. Combined volume of the gas particles is negligible.
4. Collision between particles are completely elastic.
5. The average kinetic energy of the particle is directly proportional to absolute temperature.

Step 2: A diagram to represent the Kinetic theory of gases:

Step 3: To prove

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

Step 4: Proof

We know, kinetic energy of a gas molecule, $K_1=\frac{1}{2}m{v}^2$

Kinetic energy of $n$gas molecules,

$$\begin{gathered} K=\frac{1}{2}m{v}^2\times n \\ \Rightarrow K=\frac{3}{2}\times \frac{1}{3}mn{v}^2 \\ \Rightarrow K=\frac{3}{2}PV\left(\because PV=\frac{1}{3}mn{v}^2\right) \end{gathered}$$

We know, in case of $1$ mole of ideal gas, $PV=RT$

So, Kinetic energy of gas,

$$\begin{gathered} K=\frac{3}{2}PV \\ \Rightarrow K=\frac{3}{2}RT \end{gathered}$$

From the above equation, it is clear that kinetic energy is directly proportional to absolute temperature$\left(K\propto T\right)$.