An ideal gas occupies a volume of 0.05\text{ m}^3 inside a rigid container. If the gas exe… — JAMB UTME

An ideal gas occupies a volume of $0.05\text{ m}^3$ inside a rigid container. If the gas exerts a pressure of $2.4 \times 10^5\text{ N m}^{-2}$ on the walls of the container, what is the total translational kinetic energy of the gas molecules in joules?

Answer: 18000 J

Answer

The total translational kinetic energy of the gas molecules is

18000\text{ J}

According to the kinetic theory of gases, the pressure

P

of an ideal gas is related to its total translational kinetic energy

E_k

and volume

V

P = \frac{2}{3} \frac{E_k}{V}

. Rearranging for

E_k

gives

E_k = \frac{3}{2} P V

. Substituting

P = 2.4 \times 10^5\text{ N m}^{-2}

and

V = 0.05\text{ m}^3

yields

E_k = \frac{3}{2} \times 2.4 \times 10^5 \times 0.05 = 18000\text{ J}

Step-by-Step Solution

Identify the relationship between gas pressure, volume, and translational kinetic energy from kinetic theory.

P = \frac{2}{3} \left(\frac{E_k}{V}\right) \implies E_k = \frac{3}{2} P V

From kinetic theory, pressure is two-thirds of the total translational kinetic energy per unit volume.

Substitute the provided numerical values into the equation.

E_k = \frac{3}{2} \times (2.4 \times 10^5\text{ N m}^{-2}) \times (0.05\text{ m}^3)

The given values are pressure

P = 2.4 \times 10^5\text{ N m}^{-2}

and volume

V = 0.05\text{ m}^3

Evaluate the expression to determine the numerical result.

E_k = 1.5 \times 12000 = 18000\text{ J}

Multiplying the values gives the energy in Joules.

Key Concept

Relationship between pressure, volume, and total translational kinetic energy of gas molecules (

E_k = \frac{3}{2} P V

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