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Unformatted text preview: 5.6 The Kinetic Molecular Theory of Gases 209 Deriving the Ideal Gas Law
We have shown qualitatively that the assumptions of the KMT successfully account for the observed behavior of an ideal gas. We can go further. By applying the principles of physics to the assumptions of the KMT, we can in effect derive the ideal gas law. As shown in detail in Appendix 2, we can apply the definitions of velocity, momentum, force, and pressure to the collection of particles in an ideal gas and derive the following expression for pressure:
1- P = ~[nNA~mu2)] where P is the pressure of the gas, n is the number of moles of gas, NA is Avogadro's number, m is the mass of each particle, u2 is the average of the square of the velocities of the particles, and V is the volume of the container. The quantity ~mu2 represents the average kinetic energy of a gas particle. If the average kinetic energy of an individual particle is multiplied by N A' the number of particles in a mole, we get the average kinetic energy for a mole of gas particles: (KE)avg = NA(~mu2) Kinetic energy (KE) given by the equation KE = ~m(j2is the energy due to the motion of a particle. We will discuss this further in Section 6.1. Using this definition, we can rewrite the expression for pressure as or - PV
=- 3 (KE)avg The fourth postulate of the kinetic molecular theory is that the average kinetic energy of the particles in the gas sample is directly proportional to the temperature in kelvins. Thus, since (KE)avg IX T, we can write
n = -(KE) IX T 3 avg 2 or - PV
IX T Note that this expression has been derived from the assumptions of the kinetic molecular theory. How does it compare to the ideal gas law-the equation obtained from experiment? Compare the ideal gas law,
n = RT From experiment (a) (b) (c)
as the molecules (a) A balloon filled with air at room temperature. (b) The balloon is dipped into liquid nitrogen at 77 K. (c) The balloon collapses inside slow down due to the decreased temperature. Slower molecules produce a lower pressure. ...
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