GAS LAWS: Boltzmann Constant Derivation

An ideal gas is a gas that behaves as if the only significant interactions between its atoms occurs during elastic collisions. Under ideal conditions, intramolecular force interactions due to charged particles, as well as systemic losses due to entropy, are ignored. In addition to these subatomic interactions occurring within a specified quantity of space, there are standard temperatures and pressures ( STP ) within which an ideal gas’ behavior is the most practical:

PV = nRT

This is the Ideal Gas Law equation. It is important to note that the product of pressure ( P ) and volume ( V ) yields the SI unit of energy, which is the joule ( J ):

P = ( F / A )

W = Fd 

PV = ( F / A )( V )

( V / A ) = d

PV = Fd = Work ( J )

The study of ideal gas behavior is ultimately the study of average kinetic energy of an ideal gas at STP. Whenever kinetic energy is studied, an account must also be made for the amount of mass present within a system of interest. If the number of moles ( n ) of a gas at STP remains constant, a constant of proportionality ( R ) can be obtained. This constant relates the initial and final states of a system:

( PV / nT ) = R

R = 8.31 J / mol*K

The existence of such a constant proves that changes within the initial parameters of a system will give rise to proportional changes in the final system at hand. This must be true in order for the tenets of the Law of Conservation of Energy to remain valid. If we know the temperature and quantity of an ideal gas, we can quantify the kinetic energy it possesses, and if need be, we can relate the kinetic energy to systemic changes in pressure and volume. Furthermore, the number of moles within a system can be expressed in terms that relate Avogadro’s Number ( N ) and the number of molecules within such a system:

N_o = ( 6.02 x 10²³ molecules / mole )

n = ( # moles )

nN_o = # molecules ( N )

n = ( N / N_o )

Substitution of the lattermost expression into the Ideal Gas Law expression yields the following result:

PV = nRT

PV = ( N / N_o )( RT )

PV = ( N )( R / N_o )( T )

( R / N_o ) = k_B

k_B = 1.38 x 10^-23 J / K

And,

PV = Nk_BT

We now have an expression that conveniently relates the number of molecules of an ideal gas and a standard temperature to the average kinetic energy of the system at hand.