Question

Define RMS velocity. Calculate the RMS velocity of \[C{{O}_{2}}\] at \[{{27}^{\circ }}C\].

Answer 1

Hint: For calculating the RMS, there are some factors to be calculated like the kinetic energy of the molecules, density of the molecule, molecular mass, etc. The ideal gas equation i.e., PV=RT is also used.

Complete step by step answer:
The molecules are moving in a different direction with different velocity colliding with one another as well as with the walls of the container. As a result, their individual velocity and hence the kinetic energies keep on changing even at the same temperature. However, it is found that at a particular temperature, the average kinetic energy of the gas remains constant.
At a particular temperature, if\[{{n}_{1}}\] molecules have velocity\[{{v}_{1}}\] , \[{{n}_{2}}\] molecules have velocity\[{{v}_{2}}\], \[{{n}_{3}}\]molecules have velocity\[{{v}_{3}}\] and so on, then the total kinetic energy (\[{{E}_{K}}\]) of the gas at this temperature will be:
\[{{E}_{K}}={{n}_{1}}(\dfrac{1}{2}m{{v}^{2}}_{1})+{{n}_{2}}(\dfrac{1}{2}m{{v}^{2}}_{2})+{{n}_{3}}(\dfrac{1}{2}m{{v}^{2}}_{3})........\]
\[=\dfrac{1}{2}m({{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......)\]
Where m is the mass is the molecule of the gas.
Dividing by total number molecules, average kinetic energy (\[{{E}_{K}}\]) of the gas will be
\[\overline{{{E}_{k}}}\]\[=\dfrac{{{E}_{k}}}{n}=\dfrac{1}{2}m\{\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}\}\]
In this equation, the expression\[\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}\] represents the mean of the squares of the velocity of different molecules and hence it is called mean square velocity.
Its square root is called root mean square (RMS) velocity and is represented by c. Thus the root means square velocity may be defined as the square root of the mean of the square of the speeds of different molecules of the gas. Mathematically,
\[c=\sqrt{\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}}\]
 There are many formulae of calculating RMS velocity:
If density of the gas is given: \[c=\sqrt{\dfrac{3P}{d}}\]
With ideal gas equation PV=RT, we have:
\[c=\sqrt{\dfrac{3PV}{M}}=\sqrt{\dfrac{3RT}{M}}\]
Now, calculating the RMS velocity of \[C{{O}_{2}}\] at \[{{27}^{\circ }}C\],
Here, we can apply: \[c=\sqrt{\dfrac{3RT}{M}}\]
R= gas constant = 8.314 J/K mol.
T= Temperature = 27 + 273 = 300 K
M= molar mass of \[C{{O}_{2}}\] = 44 gm/mol = 0.044 kg/mol
So, substituting all the values, we get:
\[\sqrt{\dfrac{3*8.314*300}{0.044}}\] \[=412.3m/s\]

Hence, the RMS velocity of \[C{{O}_{2}}\] at \[{{27}^{\circ }}C\] is 412.3m/s.

Note: You can also solve the values in the cgs system by taking the cgs units. Always convert the temperature into Kelvin form otherwise the answer will be wrong. Don't get confused between RMS, average, and most probable speed they all have a different formula.