Nₐ = 6.022 × 10²³ mol⁻¹ (Avogadro’s constant).
\[
m = \frac{M}{N_{\!A}}
\]
(e.g. for N₂, M = 28 g mol⁻¹ ⇒ m ≈ 4.65 × 10⁻²⁶ kg).
The ideal‑gas law in the two common forms is
\[
pV = nRT \qquad\text{and}\qquad pV = Nk_{\!B}T,
\]
where
| Assumption | Meaning for the model |
|---|---|
| Large number of molecules | Statistical treatment is valid. |
| Point‑like particles | Individual molecular volume is negligible compared with the container volume. |
| Random isotropic motion | No preferred direction; the three Cartesian components are statistically identical. |
| Elastic collisions | Kinetic energy is conserved in collisions with walls and between molecules. |
| Negligible intermolecular forces | Only brief forces during collisions are considered (ideal‑gas approximation). |
Consider a cubic container of side L (so V = L³) and a molecule of mass m moving only along the x-axis with speed cₓ.
\[
\Delta t = \frac{2L}{c_{x}} .
\]
\[
\Delta p = 2m c_{x} .
\]
\[
F = \frac{\Delta p}{\Delta t}
= \frac{2m c{x}}{2L/c{x}}
= \frac{m c_{x}^{2}}{L}.
\]
\[
p = \frac{F}{A}
= \frac{m c_{x}^{2}}{L^{3}}
= \frac{m c_{x}^{2}}{V}.
\]
For N identical molecules all moving only in the x-direction we replace \(c{x}^{2}\) by the average \(\langle c{x}^{2}\rangle\):
\[
pV = N m \langle c_{x}^{2}\rangle \qquad (1)
\]
In a real gas each molecule has independent velocity components \(c{x},c{y},c_{z}\). Random isotropic motion gives
\[
\langle c{x}^{2}\rangle = \langle c{y}^{2}\rangle = \langle c_{z}^{2}\rangle .
\]
The total mean‑square speed is the sum of the three orthogonal components:
\[
\langle c^{2}\rangle = \langle c{x}^{2}\rangle+\langle c{y}^{2}\rangle+\langle c_{z}^{2}\rangle
= 3\langle c_{x}^{2}\rangle .
\]
Substituting \(\langle c_{x}^{2}\rangle = \langle c^{2}\rangle/3\) into (1) yields the fundamental kinetic‑theory relation (syllabus 15.3.2):
\[
\boxed{\,pV = \frac{1}{3}\,N m \langle c^{2}\rangle\,} \qquad (2)
\]
\[
c_{\text{rms}} = \sqrt{\langle c^{2}\rangle } .
\]
Because \(\langle c^{2}\rangle\) appears directly in (2), the r.m.s. speed is the most convenient single number to characterise molecular motion.
Equating the kinetic‑theory result (2) with the ideal‑gas law \(pV = Nk_{\!B}T\) gives
\[
\frac{1}{3} N m \langle c^{2}\rangle = N k_{\!B} T .
\]
Cancel \(N\) and solve step‑by‑step:
\[
\langle c^{2}\rangle = \frac{3k_{\!B}T}{m},
\qquad
\frac{1}{2}m\langle c^{2}\rangle = \frac{3}{2}k_{\!B}T .
\]
Thus the average translational kinetic energy per molecule is a key formula the syllabus expects you to remember:
\[
\boxed{\,\langle E{\text{kin}}\rangle = \frac{1}{2}m\langle c^{2}\rangle = \frac{3}{2}k{\!B}T\,}.
\]
For a monatomic ideal gas the internal energy is purely translational:
\[
U = N\langle E{\text{kin}}\rangle = \frac{3}{2}Nk{\!B}T
= \frac{3}{2}nRT .
\]
For diatomic or polyatomic gases additional rotational (and at higher T, vibrational) contributions are added, but the translational part always remains \(\tfrac{3}{2}nRT\).
A typical laboratory exercise:
\[
\langle c^{2}\rangle = \frac{3pV}{Nm}.
\]
Find the pressure exerted by \(N = 2.5\times10^{25}\) molecules of nitrogen (\(\mathrm{N_{2}}\), \(M = 28\;\text{g mol}^{-1}\)) in a container of volume \(V = 0.010\;\text{m}^{3}\) when \(\langle c^{2}\rangle = (500\;\text{m s}^{-1})^{2}\).
\[
m = \frac{M}{N_{\!A}}
= \frac{28\times10^{-3}\;\text{kg mol}^{-1}}{6.022\times10^{23}\;\text{mol}^{-1}}
\approx 4.65\times10^{-26}\;\text{kg}.
\]
\[
p = \frac{1}{3}\frac{Nm\langle c^{2}\rangle}{V}
= \frac{1}{3}\frac{(2.5\times10^{25})(4.65\times10^{-26})(500^{2})}{0.010}
\approx 9.2\times10^{4}\;\text{Pa}.
\]
\[
c_{\text{rms}} = \sqrt{\langle c^{2}\rangle}=500\;\text{m s}^{-1}.
\]
\[
\langle E_{\text{kin}}\rangle = \tfrac12 m \langle c^{2}\rangle
= \tfrac12 (4.65\times10^{-26})(500^{2})
\approx 5.8\times10^{-21}\;\text{J}.
\]
\[
T = \frac{2\langle E{\text{kin}}\rangle}{3k{\!B}}
\approx \frac{2(5.8\times10^{-21})}{3(1.38\times10^{-23})}
\approx 280\;\text{K}.
\]
At constant volume, pressure is directly proportional to temperature:
\[
\frac{p{2}}{p{1}} = \frac{T{2}}{T{1}} .
\]
A sealed 2.0 L container holds air at 300 K and 1.0 atm. If the temperature is increased to 350 K, the new pressure is
\[
p{2}=p{1}\frac{T{2}}{T{1}}=1.0\;\text{atm}\times\frac{350}{300}=1.17\;\text{atm}.
\]
| Concept | Formula |
|---|---|
| Molecular mass | \(m = \dfrac{M}{N_{\!A}}\) |
| Ideal‑gas law | \(pV = nRT = Nk_{\!B}T\) |
| Kinetic‑theory pressure | \(pV = \dfrac13 N m\langle c^{2}\rangle\) |
| r.m.s. speed | \(c_{\text{rms}} = \sqrt{\langle c^{2}\rangle}\) |
| Average translational KE | \(\langle E{\text{kin}}\rangle = \dfrac12 m\langle c^{2}\rangle = \dfrac32 k{\!B}T\) |
| Internal energy (monatomic) | \(U = \dfrac32 nRT\) |
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