Match each physical quantity or concept from the kinetic theory of gases on the left with … — JAMB UTME

Match each physical quantity or concept from the kinetic theory of gases on the left with its corresponding microscopic description or mathematical relation on the right.

Root-mean-square speed ( $v_{\text{rms}}$ )Proportional to the square root of absolute temperature, given by $\sqrt{\frac{3kT}{m}}$
Average translational kinetic energy per molecule ( $\bar{E}_k$ )Directly proportional to absolute temperature, given by $\frac{3}{2}kT$
Gas pressure ( $P$ )Arises from microscopic rate of momentum transfer per unit wall area, given by $\frac{1}{3}\rho v_{\text{rms}}^2$
Absolute temperature ( $T$ )Macroscopic quantity directly proportional to the mean translational kinetic energy of the gas molecules

Answer

Root-mean-square speed corresponds to

\sqrt{\frac{3kT}{m}}

; Average translational kinetic energy per molecule corresponds to

\frac{3}{2}kT

; Gas pressure corresponds to

\frac{1}{3}\rho v_{\text{rms}}^2

; and Absolute temperature corresponds to the macroscopic measure proportional to mean translational kinetic energy.

Each kinetic theory quantity correctly matches its corresponding microscopic formula and definition derived from fundamental assumptions of ideal gas particle behavior.

Step-by-Step Solution

Analyze the microscopic derivation of root-mean-square speed

From kinetic theory,

\bar{E}_k = \frac{1}{2}m v_{\text{rms}}^2 = \frac{3}{2}kT

, which yields

v_{\text{rms}} = \sqrt{\frac{3kT}{m}}

This establishes the relationship between molecular speed, temperature, and mass.

Identify the relationship for average translational kinetic energy per molecule

The average translational kinetic energy per molecule is given directly by

\bar{E}_k = \frac{3}{2}kT

The mean kinetic energy per degree of freedom is

\frac{1}{2}kT

, summing to

\frac{3}{2}kT

for three translational dimensions.

Relate macroscopic gas pressure to microscopic particle collisions

Gas pressure is expressed as

P = \frac{1}{3}\rho v_{\text{rms}}^2

based on continuous elastic collisions of gas molecules with the container walls.

Pressure represents the average force exerted per unit area by molecular collisions.

Define absolute temperature in terms of molecular kinetic energy

Absolute temperature

T

is the macroscopic physical property directly proportional to the average kinetic energy of the molecules.

This provides the thermodynamic definition of temperature from kinetic theory.

Key Concept

Microscopic properties of ideal gas molecules and kinetic derivation of pressure and temperature

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