A gas sample enclosed in a container has an initial root-mean-square (r.m.s.) speed of 400… — JAMB UTME

A gas sample enclosed in a container has an initial root-mean-square (r.m.s.) speed of $400\text{ m s}^{-1}$ at a temperature of $27^\circ\text{C}$ . If the gas is heated at constant volume until its temperature reaches $927^\circ\text{C}$ , what is the new r.m.s. speed of the gas molecules?

A
$200\text{ m s}^{-1}$
$800\text{ m s}^{-1}$ Cevap
C
$1600\text{ m s}^{-1}$
D
$2400\text{ m s}^{-1}$

Cevap

The new root-mean-square speed of the gas molecules is

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

According to the kinetic theory of gases, the root-mean-square speed is directly proportional to the square root of the absolute temperature (

v_{\text{rms}} = \sqrt{3RT/M}

). Converting the temperatures to Kelvin gives

T_1 = 300\text{ K}

and

T_2 = 1200\text{ K}

. The ratio of absolute temperatures is

1200 / 300 = 4

. Taking the square root gives a factor of

2

, so the new r.m.s. speed is

400\text{ m s}^{-1} \times 2 = 800\text{ m s}^{-1}

Adım Adım Çözüm

Convert initial and final temperatures from Celsius to Kelvin

T_1 = 27 + 273 = 300\text{ K}

and

T_2 = 927 + 273 = 1200\text{ K}

Kinetic theory equations require absolute temperature in Kelvin.

Apply the relationship between r.m.s. speed and absolute temperature

v_{\text{rms}} \propto \sqrt{T} \implies \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}

The mean kinetic energy of gas molecules is directly proportional to absolute temperature.

Substitute the values and calculate the final speed

v_2

v_2 = 400 \times \sqrt{\frac{1200}{300}} = 400 \times \sqrt{4} = 400 \times 2 = 800\text{ m s}^{-1}

Evaluating the square root factor yields the updated r.m.s. speed.

Anahtar Kavram

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

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

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