B.3.1 Kinetic theory and gas behavior Notes

• The molar mass of a substance (in grams per mole) tells you the mass of one mole of that substance.
• For example, water (H₂O) has a molar mass of $18 \text{.g mol}^{-1}$l.

Actual Derivation of the Ideal Gas Law (in case you are really curious)

Although the IB syllabus does not require you to derive the ideal gas law, understanding where it comes from helps connect macroscopic properties (like pressure and temperature) to microscopic particle motion.

• Consider a box filled with gas particles, each of mass $m$, moving randomly in all directions.
• The gas has $N$ particles contained in a cubic box of volume $V=L^3$.
• Each particle moves in straight lines until it collides with a wall.
• When a particle hits a wall and bounces back elastically, its momentum changes by $\Delta p = 2mv_x$, where $v_x$ is the component of velocity perpendicular to the wall.
• The time between two successive collisions of this particle with the same wall is $\Delta t = \dfrac{2L}{v_x}$.
  
• Hence, the average force that this single particle exerts on the wall is
  \[
  F = \frac{\Delta p}{\Delta t} = \frac{m v_x^2}{L}
  \]
• If we sum this effect for all $N$ particles and divide by the wall’s area $A=L^2$, we obtain the pressure:
  \[
  p = \frac{\sum F}{A} = \frac{N m \langle v_x^2 \rangle}{V}
  \]
• Because the motion of particles is random, the average squared speed is distributed equally among the three directions:
  \[
  \langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle = \frac{1}{3}\langle v^2 \rangle
  \]
• Substituting this relation gives
  \[
  pV = \frac{1}{3} N m \langle v^2 \rangle = \frac{1}{3} N m v_{\text{rms}}^2,
  \]
  where $v_{\text{rms}}$ is the root-mean-square speed of the particles.
• Now, from kinetic theory, the average translational kinetic energy of one particle is related to temperature:
  \[
  \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T,
  \]
  where $k_B$ is the Boltzmann constant.
• Combining both results:
  \[
  pV = N k_B T
  \]
• If we write $N = n N_A$, where $n$ is the number of moles and $N_A$ is Avogadro’s number, and recall that $R = N_A k_B$, we finally obtain the ideal gas law:
  \[
  pV = nRT.
  \]
• Note that this derivation is based on the assumptions that:
  • gas particles have negligible volume
  • collisions are perfectly elastic
  • there are no intermolecular forces
• These assumptions are not fully true for real gases, but they work remarkably well for gases at low pressure and high temperature.
  

The ideal gas law breaks down under conditions of high pressure or low temperature, where intermolecular forces and particle volume become significant.