Question

Assertion (A) The density of hydrogen at STP is 0.09 kg m$^{-3}$ and $R = 8.31$ J mol$^{-1}$ K$^{-1}$, the root mean square speed of the hydrogen molecule is 1500 ms$^{-1}$.

Reason (R) At absolute zero temperature the root mean square velocity is also zero. (NCERT Pg. 250)

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (D)

A is false but R is true.

Answer 2

Assertion (A) Evaluation:

• Density Calculation:

  Using the ideal gas equation $\rho = \frac{PM}{RT}$ with $P = 101325 \text{ Pa}$, $T = 273.15 \text{ K}$, $M = 2 \times 10^{-3} \text{ kg/mol}$, and $R = 8.31 \text{ J mol}^{-1} \text{ K}^{-1}$:

  $\rho = \frac{101325 \times 2 \times 10^{-3}}{8.31 \times 273.15} \approx 0.0892 \text{ kg m}^{-3} \approx 0.09 \text{ kg m}^{-3}$ The density part is correct.

• RMS Speed Calculation:

  Using the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$:

  $v_{rms} = \sqrt{\frac{3 \times 8.31 \times 273.15}{2 \times 10^{-3}}} \approx 1845.25 \text{ m/s}$

  The assertion states $v_{rms}$ is 1500 m/s, which is incorrect.

• Conclusion: Assertion (A) is False because both parts must be true for the entire assertion to be true, and the calculated $v_{rms}$ does not match the stated value.

Reason (R) Evaluation:

• The formula for root mean square speed is $v_{rms} = \sqrt{\frac{3RT}{M}}$.
• If the temperature (T) is absolute zero (T = 0 K), then $v_{rms} = \sqrt{\frac{3R \times 0}{M}} = 0$.
• Therefore, Reason (R) is True.

Final Conclusion:

Assertion (A) is False, and Reason (R) is True.