Average Kinetic Energy of a Molecule

An important property of molecules in a gas is their average translational kinetic energy
For a gas, translational kinetic energy is defined as:

The kinetic energy of the molecules as determined by their temperature

The average translational kinetic energy of one particle in a gas can be deduced from the ideal gas equations relating pressure, volume, temperature and speed

Deducing the equation for translational kinetic energy

Recall the ideal gas equation:

$p V = N k T$

Also, recall the equation for the kinetic theory of gases:

$p V = \frac{1}{3} N m < c^{2} >$

The left-hand sides of the equations are equal to pV, therefore, they can be equated:

$\frac{1}{3} N m < c^{2} > = N k T$

Simplify the equation by:
- dividing both sides by N
- multiplying both sides by 3

$m < c^{2} > = 3 k T$

Recall the familiar kinetic energy equation from mechanics:

kinetic energy = $\frac{1}{2} m v^{2}$

In translational kinetic energy instead of v² for the velocity of one particle, <c²> is the average speed of all molecules
Multiplying both sides by ½ gives the equation for the average translational kinetic energy of one molecule of an ideal gas:

$E_{k} = \frac{1}{2} m < c^{2} > = \frac{3}{2} k T$

Where:
- E_K= kinetic energy of a molecule (J)
- m = mass of one molecule (kg)
- <c²> = mean square speed of a molecule (m² s^-2)
- k = Boltzmann constant
- T = temperature of the gas (K)
To calculate the average translational kinetic energy for all the molecules in a gas, multiply the kinetic energy by the number of molecules in the gas, N:

$E_{k - t o t a l} = \frac{3}{2} N k T$

Worked example

Helium can be treated like an ideal gas. Helium molecules have a root-mean-square (r.m.s) speed of 730 m s^-1 at a temperature of 45 ^oC.

Calculate the r.m.s speed of the molecules at a temperature of 80 ^oC.

Answer:

Step 1: Write down the equation for the average translational kinetic energy

$E_{k} = \frac{1}{2} m < c^{2} > = \frac{3}{2} k T$

Step 2: Find the relation between c_r.m.s and temperature T

Since m and k are constant, <c²> is directly proportional to T

$< c^{2} > \propto T$

Therefore, the relation between c_r.m.s and T is:

$c_{r . m . s} = \sqrt{< c^{2} >} \propto \sqrt{T}$

Step 3: Write the equation in full

$c_{r . m . s} = a \sqrt{T}$

where a is the constant of proportionality

Step 4: Calculate the constant of proportionality from values given by rearranging for a

T = 45 ^oC + 273.15 = 318.15 K

$a = \frac{c_{r . m . s}}{\sqrt{T}} = \frac{730}{\sqrt{318.15}} = 40.92 . . .$

Step 5: Calculate c_r.m.s at 80 ^oC by substituting the value of a and new value of T

$T = 80 ° C + 273.15 = 353.15 K$

$c_{r . m . s} = \frac{730}{\sqrt{318.15}} \times \sqrt{353.15} = 769 = 770 m s^{- 1}$

Examiner Tip

Keep in mind this particular equation for kinetic energy is only for one molecule in the gas. If you want to find the kinetic energy for all the molecules, remember to multiply by N, the total number of molecules.You can remember the equation through the rhyme ‘Average K.E is three-halves kT’.

Average Kinetic Energy of a Molecule (CIE A Level Physics)

Revision Note