Question

The temperature of the gas isi raised from $27^{\circ} \mathrm{C}$ to $927^{\circ} \mathrm{C}$, the root mean square velocity is:

  • A $\sqrt{927 / 27}$ times the earlier value
  • B Same as before
  • C Halved
  • D Doubled

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Answer

Correct Option:

D) Doubled

To determine how the root mean square (r.m.s.) velocity of a gas changes with temperature, we use the relation:

$$ U_{2} = U_{1} \sqrt{\frac{T_{2}}{T_{1}}} $$

Here, we need to convert the given temperatures from Celsius to Kelvin. The initial temperature $T_{1}$ is $27^\circ \text{C}$, which converts to:

$$ T_{1} = 27 + 273 = 300 , \text{K} $$

The final temperature $T_{2}$ is $927^\circ \text{C}$, which converts to:

$$ T_{2} = 927 + 273 = 1200 , \text{K} $$

Now, substituting these values into the formula:

$$ U_{2} = U_{1} \sqrt{\frac{1200}{300}} = U_{1} \sqrt{4} = U_{1} \times 2 $$

Thus, the root mean square velocity will be doubled when the temperature of the gas is raised from $27^\circ \text{C}$ to $927^\circ \text{C}$.


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