9.14: Kinetic Theory of Gases: Molecular Speeds
 Page ID
 49457
Other sections state that increasing the temperature increases the speeds at which molecules move. We are now in a position to find just how large that increase is for a gaseous substance. Combining the ideal gas law with Eq. (1) from The Total Molecular Kinetic Energy, we obtain
\(\begin{align} & PV=nRT=\tfrac{\text{1}}{\text{3}}Nm\text{(}u^{\text{2}}\text{)}_{\text{ave}}\\ & \text{or } \text{3}RT=\frac{Nm}{n}\text{(}u^{\text{2}}\text{)}_{\text{ave}} \label{1}\end{align}\)
Since N is the number of molecules and m is the mass of each molecule, Nm is the total mass of gas. Dividing total mass by amount of substance gives molar mass M: \[M=\frac{Nm}{n}\] Substituting in Eq. \(\ref{1}\), we have \(\begin{align} & \text{ 3}RT=M(u^{\text{2}})_{\text{ave}} \\ & \text{or }(u^{\text{2}})_{\text{ave}}=\frac{\text{3}RT}{M} \\ & \text{so that }u_{rms}=\sqrt{\text{(}u^{\text{2}}\text{)}_{\text{ave}}}=\sqrt{\frac{\text{3}RT}{M}}\text{ (2)} \end{align}\) The quantity u_{rms} is called the rootmeansquare (rms) velocity because it is the square root of the mean square velocity.The rms velocity is directly proportional to the square root of temperature and inversely proportional to the square root of molar mass. Thus quadrupling the temperature of a given gas doubles the rms velocity of the molecules. Doubling this average velocity doubles the number of collisions between gas molecules and the walls of a container. It also doubles the impulse of each collision. Thus the pressure quadruples. This is indicated graphically in Fig. \(\PageIndex{1}\) . Pressure is thus directly proportional to temperature, as required by GayLussac’s law.
Figure \(\PageIndex{1}\) A microscopic interpretation of GayLussac’s law. As the temperature of a gas is increased, the velocity of the molecules is also increased. More molecules hit the sides of the container, each with a greater impulse, so that the pressure increases.
The inverse proportionality between rootmeansquare velocity and the square root of molar mass means that the heavier a molecule is, the slower it moves, which is verified by the examples below
We can compare the rates of effusion or diffusion of a known gas with that of an unknown gas to determine the molar mass of the unknown gas. A convenient equation can be derived easily by considering the kinetic energy of individual molecules rather than moles of gas:
Knowing that kinetic energy is proportional to temperature, if the two gases are at the same temperature,

 \(\text{K} \text{E}_{1} = \text{K} \text{E}_{2} \) where 1 and 2 denote the two gases. Since \(KE= \frac{1}{2} m v^{2}\),
 \( \frac{1}{2} m_{1} ( u_{\text{rms, 1}} )^{2} = \frac{1}{2} m_{2} ( u_{\text{rms, 2}} )^{2}\) where m is the atomic weight in amu/average molecule, and u_{rms} is the velocity.
Dividing,
\[\frac{m_{1}}{m_{2}} = \frac{ ( u_{rms,2} )^{2} }{u_{rms,1} )^{2} } \]
Example \(\PageIndex{1}\) : Molar Mass
What is the molar mass of an unknown gas if the gas effuses through a pinhole into a vacuum at a rate of 2 mL/min, and H_{2} effuses at 11 mL/min. Assume that the rate of effusion is proportional to the gas molecule velocities.
Solution
\[\frac{m_{1}}{m_{2}} = \frac{ ( u_{rms,2} )^{2} }{u_{rms,1} )^{2} } \\ \frac{4}{m_{2}} = \frac{2^{2}}{11^{2}} \\ m_{2} = 121 \]
Example \(\PageIndex{2}\) : RMS Velocity
Find the rms velocity for (a) H_{2} and (b) O_{2} molecules at 27°C.
Solution This problem is much easier to solve if we use SI units. Thus we choose
R = 8.314 J mol^{–1} K^{–1} = 8.314 kg m^{2}s^{–2} mol^{–1} K^{–1}
a) For H_{2} \(\begin{align}u_{\text{rms}}=\sqrt{\frac{\text{3}RT}{M}} & =\sqrt{\frac{\text{3 }\times \text{ 8}\text{.314 J mol}^{\text{1}}\text{ K}^{\text{1}}\text{ }\times \text{ 300 K}}{\text{2}\text{.016 g mol}^{\text{1}}}}\\ & =\sqrt{\text{3}\text{.712 }\times \text{ 10}^{\text{3}}\text{ }\frac{\text{kg m}^{\text{2}}\text{s}^{\text{2}}}{\text{g}}}\\ & =\sqrt{\text{3}\text{.712 }\times \text{ 10}^{\text{3}}\text{ }\times \text{ 10}^{\text{3}}\text{ }\frac{\text{g m}^{\text{2}}\text{s}^{\text{2}}}{\text{g}}}\\ & =\sqrt{\text{3}\text{.712 }}\times \text{ 10}^{\text{3}}\text{ m s}^{\text{1}}=\text{1}\text{.927 }\times \text{ 10}^{\text{3}}\text{ m s}^{\text{1}}\end{align}\) b) For O_{2} \[u_{\text{rms}}=\sqrt{\frac{\text{3 }\times \text{ 8}\text{.314 J mol}^{\text{1}}\text{ K}^{\text{1}}\text{ }\times \text{ 300 K}}{\text{32}\text{.00 g mol}^{\text{1}}}}=\text{4}\text{.836 }\times \text{ 10}^{\text{2}}\text{ m s}^{\text{1}}\]
Now we can see the microscopic basis for Avogadro’s law. Most of the volume in H_{2}, O_{2} or any gas is empty space, and that empty space is the same for a given amount of any gas at the same temperature and pressure. This happens because the total kinetic energy of the molecules is the same for H_{2} or O_{2} or any other gas. The more energy they have, the more room the molecules can make for themselves by expanding against a constant pressure. This is illustrated in Fig. \(\PageIndex{2}\) , where equal numbers of H_{2} and O_{2} molecules occupy separate containers at the same temperature and pressure.
Figure \(\PageIndex{2}\) Avogadro’s law. Equal number of (a) O2 and (b) H2 molecules are shown in separate containers at the same P. The speedier H2 molecules make 4 times as many collisions with the walls, but each collision by one of the heavier O2 molecules is 4 times as effective. Therefore both gases push the piston up to the same height and occupy the same volume.
The volumes are seen to be the same. Because O_{2} molecules are 16 times heavier than H_{2} molecules, the average speed of H_{2} molecules is 4 times faster. H_{2} molecules therefore make 4 times as many collisions with walls. Based on mass, each collision of an H_{2} molecule with the wall has onesixteenth the effect of an O_{2} collision, but an H_{2} collision has 4 times the effect of an O_{2} collision when molecular velocity is considered. The net result is that each H_{2} collision is only onefourth as effective as an O_{2} collision. But since there are four times as many collisions, each onefourth as effective, the same pressure results. Thus the same number of O_{2} molecules as H_{2} molecules is required to occupy the same volume at the same temperature and pressure.
Contributors
Ed Vitz (Kutztown University), John W. Moore (UWMadison), Justin Shorb (Hope College), Xavier PratResina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.
Ed Vitz (Kutztown University), John W. Moore (UWMadison), Justin Shorb (Hope College), Xavier PratResina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.