Pressure, Temperature and Molecular Speeds: Quick Revision

A brief note on Pressure, Temperature and Molecular Speeds. Read it before a test. Try each question at the end before you open its answer.

The model

• A gas is a very large number of tiny molecules in random motion.
• They exert no forces on each other except in collisions, and every collision is elastic.

Pressure

• One molecule gives a wall $2mv_x$ in each hit and hits it $\dfrac{v_x}{2L}$ times each second.
• Pressure: $p = \tfrac13 nm\bar{v^2} = \tfrac13\rho\bar{v^2}$, with $n$ molecules in each cubic metre.
• In terms of energy: $p = \tfrac23\,\dfrac{E}{V}$, where $E$ is the total translational kinetic energy.

Temperature

• Total energy: $E = \tfrac32 nRT$ for $n$ moles.
• One molecule: $\bar{K} = \tfrac32 k_BT$, with $k_B = 1.38 \times 10^{-23}$ J/K.
• Temperature measures the average kinetic energy. At one temperature it is the same for every gas.

Speeds

• rms speed: $v_{rms} = \sqrt{3RT/M}$, with $M$ in kg/mol.
• Average speed: $v_{avg} = \sqrt{8RT/(\pi M)}$.
• Most probable speed: $v_{mp} = \sqrt{2RT/M}$.
• $v_{mp} : v_{avg} : v_{rms} = 1 : 1.128 : 1.225$.
• A hotter gas has a speed curve that is further right, lower and wider.

A worked example

Nitrogen ($M = 28$ g/mol) is at $290$ K. Find the average kinetic energy of a molecule and the rms speed.

1. Energy. $\bar{K} = \tfrac32 \times 1.38 \times 10^{-23}\ \text{J/K} \times 290\ \text{K} = 6.00 \times 10^{-21}$ J.
2. Speed. First $3RT = 3 \times 8.314\ \text{J/(mol K)} \times 290\ \text{K} = 7233\ \text{J/mol}$. Then $v_{rms} = \sqrt{\dfrac{7233\ \text{J/mol}}{0.028\ \text{kg/mol}}} = 508$ m/s.

Practice with new numbers

A program to try

The program works through the example above, and adds the other two speeds.

Mistakes to avoid

• Using °C in $\tfrac32 k_BT$ or $\sqrt{3RT/M}$.
• Using $M$ in g/mol in the speed formulas.
• Leaving out the one third in $p = \tfrac13\rho\bar{v^2}$.
• Doubling $T$ to double the speed. The temperature must be four times as large.

Check yourself

1. The rms speed of a gas is $400$ m/s at $200$ K. What is it at $800$ K?

2. Which has the larger average kinetic energy at $300$ K, a nitrogen molecule or a helium atom?

3. A gas has a density of $2.0\ \text{kg/m}^3$ and an rms speed of $300$ m/s. What is its pressure?

4. Why is the average speed of gas molecules larger than the most probable speed?