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General Chemistry Principles, Patterns, and Applications, 2011

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The square root of v2¯¯¯ is the root mean square (rms) speed(v rms):<br />

Equation 10.33<br />

vrms = v21+ v22 + v23+×××+ v2nn - - - -Ö<br />

Compare this with the formula used to calculate the average speed:<br />

Equation 10.34<br />

v¯= v1+ v2 + v3+×××+ vnn<br />

The rms speed <strong>and</strong> the average speed do not differ greatly (typically by less than 10%). The distinction is<br />

important, however, because the rms speed is the speed of a gas particle that has average kinetic energy.<br />

Particles of different gases at the same temperature have the same average kinetic energy, not the same<br />

average speed. In contrast, the most probable speed (vp) is the speed at which the greatest number of<br />

particles is moving. If the average kinetic energy of the particles of a gas increases linearly with increasing<br />

temperature, then tells us that the rms speed must also increase with temperature because the mass of the<br />

particles is constant. At higher temperatures, therefore, the molecules of a gas move more rapidly than at<br />

lower temperatures, <strong>and</strong> vp increases.<br />

Note the Pattern<br />

At a given temperature, all gaseous particles have the same average kinetic energy but not the same<br />

average speed.<br />

E X A M P L E 1 5<br />

The speeds of eight particles were found to be 1.0, 4.0, 4.0, 6.0, 6.0, 6.0, 8.0, <strong>and</strong> 10.0 m/s. Calculate their<br />

average speed (v¯), root mean square speed (v rms), <strong>and</strong> most probable speed (vp).<br />

Given: particle speeds<br />

Asked for: average speed (v¯), root mean square speed (v rms), <strong>and</strong> most probable speed (vp)<br />

Strategy:<br />

Use to calculate the average speed <strong>and</strong> to calculate the rms speed. Find the most probable speed by<br />

determining the speed at which the greatest number of particles is moving.<br />

Solution:<br />

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

Saylor.org<br />

940

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