A gas sample enclosed in a rigid container of fixed volume has a root-mean-square (r.m.s.)… — JAMB UTME

A gas sample enclosed in a rigid container of fixed volume has a root-mean-square (r.m.s.) speed of $500\text{ m s}^{-1}$ at a temperature of $127^\circ\text{C}$ . If the gas is heated until its pressure is quadrupled, what is the new r.m.s. speed of the gas molecules in $\text{m s}^{-1}$ ?

Answer: 1000 m s^{-1}

Answer

1000

According to kinetic theory, the pressure of a fixed volume of gas is directly proportional to its absolute temperature (

P \propto T

), and the root-mean-square speed of its molecules is proportional to the square root of absolute temperature (

v_{\text{rms}} \propto \sqrt{T}

). Quadrupling the pressure quadruples the absolute temperature in Kelvin from

400\text{ K}

1600\text{ K}

. Since the speed scales as

\sqrt{4} = 2

, the initial r.m.s. speed of

500\text{ m s}^{-1}

doubles to

1000\text{ m s}^{-1}

Step-by-Step Solution

Convert the initial temperature to absolute temperature (Kelvin)

T_1 = 127^\circ\text{C} + 273 = 400\text{ K}

Kinetic theory relationships and gas laws require temperature in absolute units (Kelvin).

Determine the new absolute temperature based on the pressure change at constant volume

T_2 = 4 \times T_1 = 1600\text{ K}

For a fixed volume of gas, pressure is directly proportional to absolute temperature (

P \propto T

). Therefore, quadrupling the pressure quadruples the absolute temperature.

Calculate the new root-mean-square speed using the square root relationship

v_2 = v_1 \sqrt{\frac{T_2}{T_1}} = 500 \times \sqrt{4} = 1000\text{ m s}^{-1}

Root-mean-square speed is proportional to the square root of absolute temperature (

v_{\text{rms}} \propto \sqrt{T}

Key Concept

Relationship between microscopic kinetic parameters (r.m.s. speed) and macroscopic state variables (pressure and absolute temperature)

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