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General Chemistry Principles, Patterns, and Applications, 2011

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KE¯¯¯¯¯¯=12M1v2rms1=12M2v2rms2<br />

Multiplying both sides by 2 <strong>and</strong> rearranging give<br />

Equation 10.37<br />

v2rms2v2rms1= M1M2<br />

Taking the square root of both sides gives<br />

Equation 10.38<br />

vrms2vrms1= M1M2 - - - -Ö<br />

Thus the rate at which a molecule, or a mole of molecules, diffuses or effuses is directly related to the speed at which it<br />

moves. shows that Graham’s law is a direct consequence of the fact that gaseous molecules at the same temperature<br />

have the same average kinetic energy.<br />

Typically, gaseous molecules have a speed of hundreds of meters per second (hundreds of miles per hour). The effect<br />

of molar mass on these speeds is dramatic, as illustrated in for some common gases. Because all gases have the same<br />

average kinetic energy, according to the Boltzmann distribution, molecules with lower masses, such as hydrogen <strong>and</strong><br />

helium, have a wider distribution of speeds. The postulates of the kinetic molecular theory of gases lead to the<br />

following equation, which directly relates molar mass, temperature, <strong>and</strong> rms speed:<br />

Equation 10.39<br />

vrms = 3RTM - - - - - Ö<br />

In this equation, vrms has units of meters per second; consequently, the units of molar mass M are kilograms per mole,<br />

temperature T is expressed in kelvins, <strong>and</strong> the ideal gas constant R has the value 8.3145 J/(K·mol).<br />

Figure 10.19 The Wide Variation in Molecular Speeds Observed at 298 K for Gases with Different Molar Masses<br />

Saylor URL: http://www.saylor.org/books<br />

Saylor.org<br />

950

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