Average Molecular Kinetic Energy (AQA A Level Physics): Revision Note

Average molecular kinetic energy

• An important property of molecules in a gas is their average kinetic energy

• This can be deduced from the ideal gas equations relating pressure, volume, temperature and speed

• Recall the ideal gas equation:

$pV=NkT$

• Also, recall the equation linking pressure and mean square speed of the molecules:

$pV=\frac{1}{3}Nm{\left(c_{rms}\right)}^2$

• The left-hand sides of both equations are equal ($pV$)

• This means the right-hand sides are also equal:

$\frac{1}{3}Nm{\left(c_{rms}\right)}^2=NkT$

• $N$ will cancel out on both sides and multiplying by 3 on both sides too obtains the equation:

$m{\left(c_{rms}\right)}^2=3kT$

• Recall the familiar kinetic energy equation from mechanics:

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

• Instead of $v^2$ for the velocity of one particle, ${\left(c_{rms}\right)}^2$ is the average speed of all molecules

• Multiplying both sides of the equation by ½ obtains the average molecular kinetic energy of the molecules of an ideal gas:

$E_k=\frac{1}{2}m{\left(c_{rms}\right)}^2=\frac{3}{2}kT$

• Where:

  • $E_k$ = kinetic energy of a molecule (J)

  • $m$ = mass of one molecule (kg)

  •  ${\left(c_{rms}\right)}^2$ = mean square speed of a molecule (m² s^-2)

  • $k$ = Boltzmann constant

  • $T$ = temperature of the gas (K)

• Note: this is the average kinetic energy for only one molecule of the gas

• A key feature of this equation is that the mean kinetic energy of an ideal gas molecule is proportional to its thermodynamic temperature

$E_k\propto T$

• The Boltzmann constant $k$ can be replaced with

$k=\frac{R}{N_A}$

• Substituting this into the average molecular kinetic energy equation means it can also be written as:

$E_k=\frac{1}{2}m{\left(c_{rms}\right)}^2=\frac{3}{2}kT=\frac{3RT}{2N_A}$

Internal energy and temperature

• Recall from Ideal Gas Internal Energy that, for an ideal gas, internal energy is equal to the sum of all kinetic energies

  • This means that, for a system of $N$ particles, the internal energy $U$ is equal to:

$U=\frac{3}{2}NkT$

• This is just the kinetic energy of a single average particle, multiplied by $N$

• This equation shows that internal energy and temperature are directly proportional

$U\propto T$