# Kinetic Theory of Gases: The Total Molecular Kinetic Energy

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

(1)

The kinetic energy of an individual molecule is ½ *m* (*u*^{2})_{ave}, and so the average kinetic energy (*E _{k}*)

_{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

(*E _{k}*)

_{ave}= (½

*m*

*u*

^{2})

_{ave}= ½

*m*(

*u*

^{2})

_{ave}

The total kinetic energy *E _{k}* is just the number of molecules times this average:

*E _{k}* =

*N*× (

*E*)

_{k}_{ave}=

*N*× ½

*m*(

*u*

^{2})

_{ave}

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

Substituting from Eq.

- (2)

or (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 *E _{k}* (4)

If we divide both sides of Eq. (4) by *n* and multiply by ,

The term *E*_{k}/*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 *E _{m}* by we have

(5)

The molar kinetic energy of a gas is proportional to its temperature, and the proportionality constant is 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*.

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