Kinetic Theory of Gases: The Total Molecular Kinetic Energy

Submitted by ChemPRIME Staff on Thu, 12/16/2010 - 13:26

Equation (1) from Postulates of the Kinetic Theory can tell us a lot more than this about gases, however. If both sides are multiplied by V, we have

$PV=\tfrac{\text{1}}{\text{3}}Nm(u^{\text{2}})_{\text{ave}}$      (1)

The kinetic energy of an individual molecule is ½ m (u2)ave, and so the average kinetic energy (Ek)ave of a collection of molecules, all of the same massA measure of the force required to impart unit acceleration to an object; mass is proportional to chemical amount, which represents the quantity of matter in an object. m is

(Ek)ave = (½ m u2)ave = ½ m (u2)ave

The total kinetic energy Ek is just the number of molecules times this average:

Ek = N × (Ek)ave = N × ½ m (u2)ave

or, multiplying both sides by 3/3 (i.e., by 1)

$E_{k}=\tfrac{\text{3}}{\text{3}}\text{ }\times \text{ }\tfrac{\text{1}}{\text{2}}Nm(u^{\text{2}})_{\text{ave}}=\tfrac{\text{3}}{\text{2}}\text{ }\times \text{ }\tfrac{\text{1}}{\text{3}}Nm(u^{\text{2}})_{\text{ave}}$

Substituting from Eq.

$E_{k}=\tfrac{\text{3}}{\text{2}}PV$      (2)

or       $PV=\tfrac{\text{2}}{\text{3}}E_{k}$      (3)

The productA substance produced by a chemical reaction. of the pressureForce per unit area; in gases arising from the force exerted by collisions of gas molecules with the wall of the container. and the volume of a gas is two-thirds the total kinetic energy of the molecules of the gas. Now we can understand why PV comes out in joules—it is indeed energy. According to postulate 4 of the kinetic theory, gas molecules have constant total kinetic energy. This is reflected on the macroscopic scale by the constancy of PV, or, in other words, by Boyle's law. The kinetic theory also gives an important insight into what the temperatureA physical property that indicates whether one object can transfer thermal energy to another object. of gas means on a microscopic level. We know from the [[CoreChem:The Ideal Gas EquationAn equation which gives a simple relationship among the pressure, volume, temperature, and chemical amount (moles) of a well behaved gas: PV, = nRT.|ideal gasA hypothetical gas for which the relationship among the pressure, volume, temperature, and chemical amount (moles) can be described by simple proportionalities summarized by the ideal gas equation, PV = nRT. law]] that PV = nRT. Substituting this into Eq. (3),

nRT = ⅔ Ek      (4)

If we divide both sides of Eq. (4) by n and multiply by $\tfrac{3}{2}$,

$\frac{E_{k}}{n}=\tfrac{\text{3}}{2}RT$

The term Ek/n is the total kinetic energy divided by the amount of substanceA material that is either an element or that has a fixed ratio of elements in its chemical formula., that is, the molar kinetic energy. Representing molar kinetic energy Em by we have

$E_{\text{m}}=\tfrac{\text{3}}{2}RT$      (5)

The molar kinetic energy of a gas is proportional to its temperature, and the proportionality constant is $\tfrac{3}{2}$ times the gas constantA proportionality constant between the product of the pressure and volume of a gas and the product of the chemical amount (moles) and temperature of the gas. R.