9.5 The kinetic-molecular theory  (Page 2/6)

Molecular velocities and kinetic energy

The previous discussion showed that the KMT qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws. To do this, we must first look at velocities and kinetic energies of gas molecules, and the temperature of a gas sample.

In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed ( [link] ).

The kinetic energy (KE) of a particle of mass ( m ) and speed ( u ) is given by:

Expressing mass in kilograms and speed in meters per second will yield energy values in units of joules (J = kg m ² s ^–2 ). To deal with a large number of gas molecules, we use averages for both speed and kinetic energy. In the KMT, the root mean square velocity of a particle, u _rms , is defined as the square root of the average of the squares of the velocities with n = the number of particles:

The average kinetic energy, KE _avg , is then equal to:

The KE _avg of a collection of gas molecules is also directly proportional to the temperature of the gas and may be described by the equation:

where R is the gas constant and T is the kelvin temperature. When used in this equation, the appropriate form of the gas constant is 8.314 J/K (8.314 kg m ² s ^–2 K ^–1 ). These two separate equations for KE _avg may be combined and rearranged to yield a relation between molecular speed and temperature: