Deriving the Kinetic Theory Equation Thermal Physics Lesson 6

Learning Objectives Derive the kinetic theory equation.

Here we go… We consider a molecule of mass m in a box of dimensions l x, l y and l z with a velocity components u 1, v 1 and w 1 in the x, y and z directions respectively.

A derivation of two halves… The strategy is to derive the pressure for one molecule on one face (1 st half) and then sum all the pressures from all the molecules (2 nd half). The first part involves Newton’s second law which states that the force on a body is equal to the rate of change of momentum.

Also note… The speed c of the molecule is given by:- This comes from doing Pythagoras’ theorem in 3 dimensions. We will use this later in the 2 nd half

Find the force on one molecule… When the molecule impacts with Face A the x- component of its momentum is changed from mu 1 to -mu 1. But we want the rate of change of momentum, so need to divide by the time taken between collisions with Face A.

Time, t, between collisions… So Newton’s 2 nd Law gives us:- And Newton’s 3 rd Law gives us:-

Pressure Recall that to work out the pressure:- So the pressure p 1 on face A:- So that’s the first bit done! – the pressure from one molecule in one direction.

Summing the pressures The total pressure on face A can be calculated by summing the pressures of all of the molecules:- Where p 2, p 3… refer to the pressures of all the other molecules up to N molecules.

Summing the pressures The last line can be rewritten as:- Where the mean square x-component velocity is given by:- And similar equations can be derived for the y and z components

Almost there… But we want our equation to include the root mean square speed in all directions. Using Pythagoras’s theorem, the speed for one molecule is given by:- You can show that the root mean square speed for all the molecules is:-

Almost there… Because the motions are random we can write:- Otherwise there would be a drift of particles in one direction. So using the above two equations:- So now we can write:-

Final Rearrangement The Kinetic Theory Equation:- Can also be re-written as:- Because:-

Recap Use Newton’s Second Law Use Newton’s Third Law Calculate pressure due to one molecule (force/area) Sum the pressures of all the molecules. Rewrite the speed in terms of root mean square speed.

Deriving Ideal Gas Equation From Boyle’s Law: From Pressure Law: From Avogadro’s Law: Combining these three: Rewriting using the gas constant R: Therefore:-

Mean kinetic energy The mean kinetic energy of a molecule is the total kinetic energy of all the molecules/total number of molecules. The root mean square speed is defined as: …and so the mean kinetic energy of one molecule is gvien by…

Mean Kinetic Energy Note we have two equations with pV on the left hand side. (N=nN A ) The Boltzmann constant, k, is defined as R/N A.

Mean Kinetic Energy Multiply both sides by 3/2. Mean kinetic energy of a molecule of an ideal gas. Kinetic energy of one mole of gas (because R=N A k) Kinetic energy of n moles of gas.