Explain the origin of gas pressure and the way pressure changes in terms of the forces exerted by particles when they collide with surfaces – i.e. as a force per unit area (P = F/A).
The particle model uses the term particle to mean any microscopic entity – an atom, a molecule or an ion – that behaves as a single unit in collisions. In a mono‑atomic gas (e.g. He) each particle is an atom; in a di‑atomic gas (e.g. O₂) each particle is a molecule; in an ionised gas (plasma) each particle may be an ion.
The average translational kinetic energy of a particle is proportional to the absolute temperature:
\[
\langle Ek\rangle = \tfrac{3}{2}\,k{\rm B}\,T
\]
where \(k_{\rm B}=1.38\times10^{-23}\,\text{J K}^{-1}\) is Boltzmann’s constant. As temperature rises, particles move faster and collide with the walls more energetically.
When a gas particle strikes a surface it exerts a force on that surface. The cumulative effect of many collisions over an area \(A\) gives the pressure:
\[
P = \frac{F}{A}
\]
\[
\Delta p = mv - (-mv) = 2mv .
\]
\[
F_{\text{particle}} = \frac{\Delta p}{\Delta t}.
\]
\[
F = \sum{i=1}^{N{\text{coll}}} \frac{\Delta pi}{\Delta ti}.
\]
\[
P = \frac{F}{A} = \frac{1}{A}\sum \frac{\Delta pi}{\Delta ti}.
\]
In real gases particles exert short‑range attractive or repulsive forces. The magnitude of these forces depends on the distance between particles:
These interactions explain why real gases deviate from the ideal‑gas equation at high pressures (particles forced close together) or low temperatures (attractive forces become significant).
| Variable | What changes at the particle level? | Resulting effect on pressure |
|---|---|---|
| Temperature \(T\) | Average speed ↑ → momentum change per collision ↑ and collision frequency ↑ | Force ↑ → pressure ↑ (Gay‑Lussac’s law) |
| Volume \(V\) | Distance between wall and particle ↓ → particles reach the wall more often (collision frequency ↑) | Force ↑ → pressure ↑ (Boyle’s law) |
| Amount of gas \(n\) | Number of particles ↑ → more collisions per unit time | Force ↑ → pressure ↑ (direct proportionality) |
| Nature of the gas | Particle size & intermolecular forces differ → affect how closely particles can be packed and how much kinetic energy is lost in collisions | Real‑gas deviations from ideal behaviour at high \(p\) or low \(T\) |
\[
pV = \text{constant}
\]
Halving the volume halves the average distance a particle travels before hitting a wall, so the collision frequency doubles; the average force per unit area therefore doubles, giving twice the pressure.
\[
\frac{V1}{T1} = \frac{V2}{T2}
\]
Raising the temperature increases the average speed of the particles. To keep the pressure unchanged, the container must expand so that the collision frequency per unit area remains the same.
\[
\frac{p1}{T1} = \frac{p2}{T2}
\]
At a fixed volume, a higher temperature means faster particles and a larger momentum change per collision, so the pressure rises in direct proportion to \(T\).
\[
\frac{pV}{T} = \text{constant}
\]
\[
pV = nRT
\]
Derivable from the particle model by substituting \(\langle Ek\rangle = \tfrac{3}{2}k{\rm B}T\) into the kinetic‑theory expression for pressure:
\[
p = \frac{1}{3}\frac{N}{V}m\langle v^2\rangle,
\]
where \(N = nN{\!A}\) and \(R = N{\!A}k_{\rm B}\).
A 2 L container holds 0.5 mol of an ideal gas at 300 K. It is heated to 600 K while the volume remains fixed.
Using \(p1/T1 = p2/T2\):
\[
p2 = p1\frac{600}{300}=2p_1.
\]
The pressure doubles because the average particle speed (and thus momentum change per collision) doubles.
The same gas is now compressed isothermally from 2 L to 1 L.
From Boyle’s law, \(p1V1 = p2V2\) → \(p2 = p1\frac{2}{1}=2p_1\).
Halving the volume halves the mean free path, doubling the collision frequency.
Adding another 0.5 mol (doubling \(n\)) while keeping \(V\) and \(T\) unchanged gives, from the ideal‑gas equation,
\[
p2 = 2p1.
\]
Twice as many particles mean twice as many collisions per second.
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