Ideal Gas Law & Maxwell-Boltzmann Distribution (Lecture Notes)

Ideal Gas Law and Related Concepts

• Start from empirical gas laws: Boyle's, Charles's, and Avogadro's laws lead to a relationship where volume and pressure are connected to the number of moles and temperature: PV ∝ nT.
  • To turn the proportionality into an equality, introduce a constant: $PV = nRT.$
  • Here, R is the ideal gas constant, determined experimentally; its value depends on the chosen units for P, V, and T. Examples:
  • In SI units: $R \,=\, 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}$
  • In L·atm units: $R \,=\, 0.082057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}$
  • Boltzmann constant kB relates microscopic and macroscopic scales: $k</em>B = 1.38 \times 10^{-23}\ \mathrm{J\,K^{-1}}.$ The macroscopic constant R is related to kB by $R = N</em>A k_B.$
• Standard pressure conventions and molar quantities
  • The standard pressure convention: a circle drawn on a variable in diagrams often indicates 1 bar of pressure (standard pressure for the context).
  • Molar volume: $V_m = \frac{V}{n}$ (volume per mole).
  • Molar enthalpy and molar entropy: $H<em>m = \frac{H}{n},\quad S</em>m = \frac{S}{n}.$
• Standard states: STP vs SATP
  • STP (older standard): $T = 0^{\circ}\mathrm{C} = 273.15\ \mathrm{K},\quad P = 1\ \mathrm{atm}.$
  • SATP (more common nowadays): $T = 25^{\circ}\mathrm{C} = 298.15\ \mathrm{K},\quad P = 1\ \mathrm{atm}.$
  • Molar volumes at these conditions:
  • At STP: $V_m \approx 22.414\ \mathrm{L\,mol^{-1}}.$
  • At SATP: $V_m \approx 24.465\ \mathrm{L\,mol^{-1}}.$
• Gas mixtures and Dalton’s law (partial pressures)
  • In a container with multiple ideal gases, the total pressure is the sum of the individual (partial) pressures:
  • Partial pressure of component j: $P<em>j = \frac{n</em>j RT}{V}.$ (Ideal gas law applied to each component.)
  • Total moles: $n = \sum<em>j n</em>j.$ (moles add up in the mixture)
  • Mole fractions: $x<em>j = \frac{n</em>j}{n},\quad \sum<em>j x</em>j = 1.$
  • Total pressure: $P = \sum<em>j P</em>j.$ (Dalton’s law of partial pressures)
  • Equivalently, $P<em>j = x</em>j P.$
• Consistency checks for mixtures
  • Sum of mole fractions equals one: $\sum<em>j x</em>j = 1.$
  • Sum of partial pressures equals total pressure: $P = \sum<em>j P</em>j.$
• Conceptual picture: ideal gas as kinetic theory toy model
  • Assumptions for the ideal gas model (molecules):
  • Point particles with mass, in ceaseless random motion, obeying classical mechanics (momentum, kinetic energy).
  • Collisions are elastic; there are no intermolecular potential energies that affect the motion in the ideal gas approximation.
  • The total pressure arises from momentum transfer of many molecules colliding with container walls.
  • In real gases, interactions (London dispersion, dipole-dipole, hydrogen bonding) introduce potential energy and deviations from ideal behavior.
  • The mixture example: each gas contributes to total pressure; the total pressure is the sum of the contributions from each gas if the ideal gas assumptions hold.
• From macroscopic to microscopic view: a preview of the Maxwell–Boltzmann framework
  • The macroscopic quantities we measure (P, T) are connected to microscopic molecular speeds, masses, and collisions with container walls.
  • The goal is to derive an equation of state from kinetic theory by describing the speed distribution of molecules and relating it to pressure.
  • Outline of the pathway: speed -> momentum -> force -> pressure.
• Maxwell–Boltzmann distribution of speeds (topic 1b)
  • Intuition: molecules have a distribution of speeds; some slow, some fast, with most around a most probable speed depending on temperature.
  • Define a probability density for speeds: the area under the curve between v1 and v2 gives the probability of finding a molecule with speed in that range.
  • Boltzmann distribution of energy (foundation): for energy E, the probability density is proportional to $e^{-E/(k<em>B T)}.$ For kinetic energy, $E = \tfrac{1}{2} m v^2,$ so the energy distribution is proportional to $e^{-\frac{m v^2}{2 k</em>B T}}.$
  • Step 1: distribution for a single velocity component (e.g., along x) is Gaussian
  • $f(v<em>x) = \sqrt{\frac{m}{2 \pi k</em>B T}}\; e^{-\frac{m v<em>x^2}{2 k</em>B T}}.$
  • This distribution is normalized: $\int<em>{-\infty}^{\infty} f(v</em>x)\, dv_x = 1.$
  • Step 2: combine x, y, z components to get a distribution in 3D velocity space
  • The joint distribution in vector form is
    • $f(\mathbf{v}) = \left(\frac{m}{2 \pi k<em>B T}\right)^{3/2} e^{-\frac{m(v</em>x^2 + v<em>y^2 + v</em>z^2)}{2 k_B T}}.$
  • Transition from velocity components to speed: in velocity space, the volume element is $d^3 v = dv<em>x \, dv</em>y \, dv_z.$ The transformation to a radial speed variable uses spherical coordinates, where the angular integration gives a factor of 4π and the radial part is 4π v^2 dv, i.e. $d^3 v = 4\pi v^2 \, dv$ when integrating over directions and focusing on the speed magnitude v.
  • Step 3: obtain the speed distribution f(v) for v ≥ 0 by integrating over directions (angle integration) and normalizing
  • Resulting Maxwell–Boltzmann speed distribution:
    • $f(v) = 4\pi \left(\frac{m}{2 \pi k<em>B T}\right)^{3/2} v^2 \exp\left(-\frac{m v^2}{2 k</em>B T}\right), \quad v \ge 0.$
  • Normalization condition: $\int_{0}^{\infty} f(v)\, dv = 1.$
  • Notes about normalization and constants
  • The Gaussian form for each velocity component and the Jacobian for the transformation to speed lead to the prefactor $\left(\frac{m}{2 \pi k_B T}\right)^{3/2}$ and the extra factor of $4\pi v^2$ for the speed distribution.
  • The distribution for a single component is Gaussian: a bell-shaped curve with width set by temperature and mass; higher T broadens the distribution, higher m narrows it.
  • Physical interpretation and implications
  • The MB distribution provides the statistical distribution of molecule speeds in an ideal gas at temperature T.
  • It is consistent with the kinetic theory picture where pressure arises from repeated molecular collisions with container walls.
  • Most probable speed, mean speed, and root-mean-square speed depend on T and m, e.g. the rms speed scales as $v<em>{\mathrm{rms}} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k</em>B T}{m}}.$ (Note: this exact relation is a standard MB result; the lecture notes indicate the direction of deriving such relationships.)
  • Practical caveats
  • The lecture describes the derivation as an “ugly derivation” and notes the expectation to finish it in a subsequent session.
  • The ultimate aim is to connect speeds to momentum, momentum to forces, and forces to pressure, culminating in a molecular-level understanding of the ideal gas law.
• Quick takeaways and connections
  • The ideal gas law PV = nRT marries macroscopic observables with a microscopic constant R that encapsulates molecular properties (via NA and kB).
  • In mixtures, Dalton’s law allows us to decompose pressure into partial pressures with the relationships between nj, xj, and P.
  • The molar quantities (Vm, Hm, S_m) provide a per-mole perspective that often simplifies comparisons between gases.
  • The Maxwell–Boltzmann distribution provides the probabilistic backbone for predicting how molecular speeds contribute to macroscopic properties like pressure in kinetic theory.
• Practical implications for exam preparation
  • Be able to derive and manipulate: $PV = nRT,$ $P<em>j = \frac{n</em>j RT}{V},\quad P = \sum<em>j P</em>j,\quad x<em>j = \frac{n</em>j}{n},\quad \sum<em>j x</em>j = 1.$
  • Recall standard states and molar quantities, including typical numerical values for R and kB and the relationships $R = N</em>A k_B.$
  • Understand the logic behind the Maxwell–Boltzmann distributions for both velocity components and speed, and the role of the normalization integrals in determining the prefactors.