Chapter 21, Problem 1OQ

To determine

If the molecules in the two cylinders have same rms speeds, find the false statements.

Answer to Problem 1OQ

Option (c)

Explanation of Solution

Write the expression for rms speed of the oxygen gas molecules.

  $v_{\text{rms,1}}=\sqrt{\frac{3R{T}_1}{M_1}}$                                                                                                  (I)

Here, $v_{\text{rms,1}}$ is the rms speed of the oxygen gas molecules, $R$ is the gas constant, $T_1$ is the temperature of the oxygen gas, and $M_1$ is the molar mass of the oxygen gas.

Similarly the expression for rms speed of the nitrogen gas molecules

  $v_{\text{rms,2}}=\sqrt{\frac{3R{T}_2}{M_2}}$                                                                                                   (II)

Here, $v_{\text{rms,2}}$ is the rms speed of the nitrogen gas molecules, $T_2$ is the temperature of the nitrogen gas, and $M_2$ is the molar mass of the nitrogen gas.

Molecules in both the cylinders have same rms speed.

Equate (I) and (II).

  $\begin{array}{c}\sqrt{\frac{3R{T}_1}{M_1}}=\sqrt{\frac{3R{T}_2}{M_2}}\\ \frac{3R{T}_1}{M_1}=\frac{3R{T}_2}{M_2}\\ \frac{T_1}{M_1}=\frac{T_2}{M_2}\end{array}$                                                                                               (III)

Conclusion:

Rearrange the equation (III) for $T_1$.

  $T_1=\frac{M_1}{M_2}{T}_2$

If mass of the nitrogen gas is less than the oxygen gas ($M_2<M_1$) so the temperature $T_2$ must be less than the temperature $T_1$ to keep the ratio same, so the option (c) is False.

If the above ratio is keep same as the molar mass of the two different gases, meanwhile temperature is also different. Therefore, option (a) is True.

If mass of the nitrogen gas is greater than the oxygen gas ($M_2>M_1$) so the temperature $T_2$ must be greater than the temperature $T_1$ to keep the ratio same, so the option (b) is True.

Average kinetic energy depend on temperature so if $T_2$ is less than the $T_1$, then the kinetic energy for oxygen is also be less than kinetic energy for nitrogen gas so the option (d) is True.