19-3 Pressure, Temperature, and RMS Speed#

Prompts

How does kinetic theory explain gas pressure? What role do molecular collisions play?
Write the relation between pressure \(p\) and rms speed \(v_{\text{rms}}\). How does \(p\) depend on density \(\rho\)?
Derive or state \(v_{\text{rms}} = \sqrt{3RT/M}\). At the same temperature, which gas has higher rms speed: H\(_2\) or O\(_2\)?
What is the average translational kinetic energy per molecule for an ideal gas? How does it depend on temperature?

Lecture Notes#

Overview#

Pressure arises from molecular collisions with the container walls—momentum transfer per unit area per unit time.
Kinetic theory connects pressure and temperature to the rms speed \(v_{\text{rms}}\) of the molecules.
Temperature is a measure of average translational kinetic energy: \(K_{\text{avg}} = \frac{3}{2}kT\).

Pressure and rms speed#

For an ideal gas, pressure is related to the root-mean-square speed:

(173)#\[ p = \frac{nM\,v_{\text{rms}}^2}{3V} = \frac{1}{3}\rho\,v_{\text{rms}}^2 \]

\(v_{\text{rms}} = \sqrt{(v^2)_{\text{avg}}}\)—square each speed, average, then take the square root.
\(\rho = nM/V\)—density.

RMS speed in terms of temperature#

Combining the pressure relation with the ideal gas law \(pV = nRT\):

(174)#\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}} \]

\(M\): molar mass; \(m\): mass per molecule.
Lighter molecules move faster at the same \(T\): \(v_{\text{rms}} \propto 1/\sqrt{M}\).

Gas	\(v_{\text{rms}}\) at 300 K (m/s)
H\(_2\)	~1920
N\(_2\)	~517
O\(_2\)	~483

Average kinetic energy and temperature#

The average translational kinetic energy per molecule is

(175)#\[ K_{\text{avg}} = \frac{1}{2}m\,v_{\text{rms}}^2 = \frac{3}{2}kT \]

\(K_{\text{avg}}\) depends only on \(T\)—not on mass or pressure. At the same temperature, all ideal gases have the same \(K_{\text{avg}}\).
Pressure in terms of \(K_{\text{avg}}\): \(p = \frac{2}{3}(N/V)K_{\text{avg}}\).

Temperature as kinetic energy

Temperature measures the average translational kinetic energy of molecules. Higher \(T\) → faster molecules → higher \(v_{\text{rms}}\) and \(K_{\text{avg}}\).

Poll: RMS speed

At the same temperature, the rms speed of O\(_2\) molecules is ___ the rms speed of H\(_2\) molecules. (Molar mass: O\(_2\) = 32 g/mol, H\(_2\) = 2 g/mol)

(A) Greater than
(B) Less than
(C) Equal to

Summary#

\(p = \frac{1}{3}\rho\,v_{\text{rms}}^2\)—pressure from molecular collisions.
\(v_{\text{rms}} = \sqrt{3RT/M}\)—lighter molecules faster at same \(T\).
\(K_{\text{avg}} = \frac{3}{2}kT\)—temperature measures average translational KE.