Question

The RMS velocity of gas molecules at NTP cannot be calculated from which one of the following formula?
A) \[\sqrt {\dfrac{{3P}}{d}} \]
 B)\[\sqrt {\dfrac{{3PV}}{M}} \]
 C)\[\sqrt {\dfrac{{3RT}}{M}} \]
 D)\[\sqrt {\dfrac{{3RT}}{d}} \]

Answer 1

Hint: RMS velocity has a full form Root Mean Square velocity which is defined as the square root of the average of squares of different velocities of the gas molecules. We can derive its formula with the help of density which is mass by volume and ideal gas equation which is PV=nRT where n are the number of moles of gas.

Complete step-by-step answer:
In order to calculate RMS velocity, let us start with kinetic gas equation. We know that \[PV = \dfrac{1}{3}MN{u^2}\] is the kinetic gas equation in which P is the pressure of the gas, V is the volume of the gas, M is the mass of 1 molecule of gas, N is the number of the gas molecules and u is the root mean square velocity of the molecules.
We can also write the kinetic energy formula as follows:
  \[K.E. = \dfrac{1}{2}M{u^2}\] , where M is the mass of a gas molecule and u is the velocity of the gas molecule. So, we can write the kinetic gas equation for one mole of a gas when \[N = 1\] .
  \[PV = \dfrac{1}{3}M{u^2}\]
We are familiar with the ideal gas equation and we can write it for one mole of a gas as
PV= RT
Now, let us put the value of PV from kinetic gas equation to the ideal gas equation. We get,
 \[RT = \dfrac{1}{3}M{u^2}\]
 \[\begin{gathered}
   \Rightarrow 3RT = M{u^2} \\
   \Rightarrow \dfrac{{3RT}}{M} = {u^2} \\
   \Rightarrow \sqrt {\dfrac{{3RT}}{M}} = u \\
\end{gathered} \]
From this formula we can easily derive RMS velocity so, option (C) is not the correct answer.
From the ideal gas equation, which is PV=RT for 1 mole of gas, let us substitute RT in the above formula with PV since they are equal. Now, the formula becomes \[\sqrt {\dfrac{{3PV}}{M}} \] . So, (B) is also incorrect as per the question.
We know that, density is mass by volume. Therefore, in the above formula we can substitute \[\dfrac{V}{M}\] by d. Now, the formula becomes \[\sqrt {\dfrac{{3P}}{d}} \] . So, this option is again incorrect as per question.
We are left with only option (D) which is the correct answer because we cannot derive the formula of RMS velocity from it. This is because from this formula, we get the information that density is equal to mass on comparing it with the formula of option C and density is not equal to mass but is equal to mass by volume.
Hence, the correct option is (C).

Note:RMS velocity depends only on the temperature of the gas and is independent of pressure, volume, and the nature of gas for an ideal equation. This relation of temperature and average energy of molecules shows complete consistency between kinetic theory of gases and ideal gas equation along with other laws based on it.