Verified Solution • 2025–26 Session moderate

QPages 34-37: Q18: HC Verma: Concepts of Physics - Volume 2 Kinetic Theory of Gases Pages 34-37: Q18 Solution: At what temperature the mean speed of the molecules of hydrogen …

⚠️ Common Exam Traps & Conceptual Pitfalls

Students frequently substitute the rms speed formula $v_{\rm rms}=\sqrt{3RT/M}$ for the mean speed $v_{\rm mean}=\sqrt{8RT/(\pi M)}$, giving a temperature that is too low by a factor $\sqrt{3\pi/8}$. Remember to use the mean‑speed expression and the molar mass $M$ in kg mol$^{-1}$ (for H$_2$, $M=2\times10^{-3}$ kg mol$^{-1}$).

At what temperature the mean speed of the molecules of hydrogen gas equals the escape speed from the earth? Use

R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}

Step-by-Step Verified Solution

**Answer:**

T = 1.0 \times 10^4 \text{ K}

**Step-by-Step Explanation:** Escape speed from Earth

= 11.2 \times 10^3

m/s. For hydrogen,

M = 2 \times 10^{-3}

kg/mol. Mean speed

= \sqrt{8RT/\pi M}

. Setting equal to escape speed:

T = \frac{\pi M v_e^2}{8R} = \frac{\pi \times 0.002 \times (11.2 \times 10^3)^2}{8 \times 8.314} = 1.0 \times 10^4 \text{ K}

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