The root mean square speed of molecules of nitrogen gas at 27^ C… — Solution | JEE Physics

MCQEasyJEE 2023Molecular Speeds (rms, Average, Most Probable)

The root mean square speed of molecules of nitrogen gas at $27^\circ \text{C}$ is approximately : (Provided that mass of a nitrogen molecule $= 4.6 \times 10^{-26} \, \text{kg}$ and take Boltzmann constant $k_B = 1.4 \times 10^{-23} \text{J} \text{K}^{-1}$ )

A
$1260 \, \text{m/s}$
B
$91 \, \text{m/s}$
C
$523 \, \text{m/s}$
D
$27.4 \, \text{m/s}$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Temperature $T = 27^\circ \text{C} = 300 \, \text{K}$ , mass of one nitrogen molecule $m = 4.6 \times 10^{-26} \, \text{kg}$ , and Boltzmann constant $k_B = 1.4 \times 10^{-23} \, \text{J/K}$ .

Find: The root mean square speed $V_{\text{rms}}$ of nitrogen molecules.

For an ideal gas molecule,

V_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}

Substituting the given values,

V_{\text{rms}} = \sqrt{\frac{3 \times 1.4 \times 10^{-23} \times 300}{4.6 \times 10^{-26}}}

Simplifying the numerator,

3 \times 1.4 \times 300 = 1260 \times 10^{-23} = 1.26 \times 10^{-20}

So,

\frac{1.26 \times 10^{-20}}{4.6 \times 10^{-26}} = 2.73 \times 10^5

Taking the square root,

V_{\text{rms}} = \sqrt{2.73 \times 10^5} \approx 523 \, \text{m/s}

Therefore, the root mean square speed of nitrogen molecules is approximately $523 \, \text{m/s}$ . The solution working gives the correct option as C, even though the solution incorrectly states D.

Common mistakes

Using temperature directly as $27$ instead of converting it to absolute temperature is incorrect because kinetic theory formulas require Kelvin. Convert $27^\circ \text{C}$ to $300 \, \text{K}$ before substitution.
Using the wrong speed formula, such as average speed or most probable speed, gives a different numerical value. Here the required formula is specifically $V_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}$ .
Mishandling powers of $10$ during simplification can produce answers like $1260$ or $27.4$ . First combine the scientific notation carefully, then take the square root only at the final step.

Practice more Molecular Speeds (rms, Average, Most Probable) questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Molecular Speeds (rms, Average, Most Probable)

Answer

Step-by-step solution

Common mistakes

Practice more Molecular Speeds (rms, Average, Most Probable) questions

Related questions