Maxwell-Boltzmann Distribution and Kinetic Theory of Pressure

Logistics and context

  • Reading assignments: not graded; you’re responsible for doing the work on time.
  • Canvas: assignments posted with which textbook readings or sections to cover; aim to read accordingly so everyone is on the same page.
  • Quick class flow: finish the last derivation, then derive pressure from a microscopic (kinetic theory) model.
  • Quick recap from last session: derived the perfect gas law concepts and identified the gas constant as R; connected Boltzmann distribution to molecular kinetic energy to obtain the speed distribution along a single coordinate and then extended to all three coordinates.
  • Big picture: connect microscopic molecular motion to macroscopic pressure via momentum transfer to container walls.

Maxwell–Boltzmann distribution of speeds

  • Starting point: use Boltzmann distribution for kinetic energy, $K = \tfrac{1}{2} m v^2$, and integrate over the velocity components to get the speed distribution.
  • Resulting speed distribution (Maxwell–Boltzmann form) for speed $v$:
    f(v)=4π(m2πkT)3/2v2exp⁡(−mv22kT)(v≥0)f(v) = 4 \pi \left(\frac{m}{2\pi k T}\right)^{3/2} v^{2} \exp\left(-\frac{m v^{2}}{2 k T}\right) \quad (v \ge 0)
  • Key structure of $f(v)$ described in class:
    • A constant prefactor (for a fixed $m$ and $T$) multiplies the $v^2$ term and the exponential term.
    • The $v^2$ factor makes $f(v)$ start at 0 when $v=0$ (no speeds exactly at zero).
    • The exponential term suppresses large speeds, giving a tail that decays rapidly.
    • Overall, the function resembles a skewed Gaussian: starts at 0, rises to a peak, then falls off toward zero as $v \to \infty$.
  • Intuition and qualitative features:
    • At a fixed temperature, heavier (larger molar mass or molecular mass $m$) particles have distributions that peak at lower speeds and tail off sooner.
    • At higher temperature, the distribution broadens and shifts toward higher speeds (higher average kinetic energy).
    • The distribution is skewed to the right (peak left of the mean).
  • Special speeds (commonly discussed):
    • Mean speed (average over the distribution):
      ⟨v⟩=8kTπm\langle v \rangle = \sqrt{\frac{8 k T}{\pi m}}
    • Most probable speed (location of the peak of $f(v)$):
      vextmp=2kTmv_{ ext{mp}} = \sqrt{\frac{2 k T}{m}}
    • Root-mean-square speed (RMS):
      vrms=⟨v2⟩=3kTmv_{\text{rms}} = \sqrt{\langle v^{2} \rangle} = \sqrt{\frac{3 k T}{m}}
    • Relationship among these for a given $m$ and $T$:
    • $\langle v^{2} \rangle = \dfrac{3 k T}{m}$
    • $v_{\text{rms}} = \sqrt{\langle v^{2} \rangle}$
  • Expressing speeds in molar form (using molar mass $M$ and gas constant $R$):
    • $v_{ ext{mp}} = \sqrt{\dfrac{2 R T}{M}}$
    • $\langle v \rangle = \sqrt{\dfrac{8 R T}{\pi M}}$
    • $v_{\text{rms}} = \sqrt{\dfrac{3 R T}{M}}$
  • Practical interpretations:
    • The distribution provides a probabilistic view of where particle speeds lie, not a single “typical” speed.
    • The most probable speed is not the average speed due to skewness.
    • The average kinetic energy per molecule is linked to temperature via $\langle K \rangle = \tfrac{3}{2} k T$, consistent with equipartition.

Breakdown of the distribution and takeaways

  • The constant term: sets the overall normalization for fixed $m$ and $T$; the variable dependence is in $v$ through the $v^{2}$ factor and the exponential.
  • The $v^{2}$ factor implies there are no particles with zero speed (probability density vanishes at $v=0$).
  • As $v$ increases, the exponential term makes $f(v)$ decay to zero for very large speeds, giving a finite most-probable region.
  • Mass dependence: increasing $m$ makes the distribution narrower and peak at a lower speed; corresponds to heavier particles moving more slowly on average.
  • Temperature dependence: increasing $T$ broadens the distribution and shifts the peak to higher speeds; higher $T$ means higher average kinetic energy.
  • Example intuition: at a fixed $T$, helium (light) has higher most-probable speed than xenon (heavy); at the same $T$, raising $T$ shifts both distributions to higher speeds.

From speeds to pressure: kinetic theory setup (three-step outline)

  • Goal: derive an expression for pressure $P$ from a microscopic model of molecular motion and wall collisions.
  • Step 1: consider a single particle of mass $m$ hitting a wall.
    • When the particle collides elastically with the wall, its momentum component normal to the wall changes from $m vx$ to $-m vx$.
    • Momentum change per collision: (\Delta p = 2 m v_x) (magnitude).
  • Step 2: extend to many particles by counting collisions in a time interval and using the density of particles.
    • Define the container width along the wall as $x$ and the wall area as $A$ (the wall has some area facing the gas).
    • Over a short time interval $\Delta t$, particles that strike the wall are those within a volume swept toward the wall: $V_{ ext{hit}} = A \cdot (v \Delta t)$ for the relevant component toward the wall.
    • The number density (number of particles per volume) is
      n=NVn = \frac{N}{V}
    • The number of molecules in the swept volume (and thus the number likely to hit the wall in $\Delta t$) is approximately
      N<em>exthit≈n⋅V</em>exthit=nAvΔtN<em>{ ext{hit}} \approx n \cdot V</em>{ ext{hit}} = n A v \Delta t
  • Step 3: relate momentum change to force and then to pressure; account for three dimensions (isotropy) and use a differential/averaging approach
    • Total momentum transfer in time $\Delta t$ is the sum over all colliding molecules: ( \Delta p{ ext{tot}} \approx \sum 2 m vx ) for all collisions in that interval.
    • The force on the wall is
      F≈ΔpexttotΔtF \approx \frac{\Delta p_{ ext{tot}}}{\Delta t}
    • Pressure is force per area, so
      P=FA  .P = \frac{F}{A} \;.
    • In three dimensions, only the normal component toward the wall contributes to pressure; averaging over all directions (isotropy) yields
      P=NV⋅m⟨v2⟩3=n⋅m⟨v2⟩3P = \frac{N}{V} \cdot \frac{m \langle v^{2} \rangle}{3} = n \cdot \frac{m \langle v^{2} \rangle}{3}
    • Using the equipartition-like relation for MB speeds, ( \langle v^{2} \rangle = \dfrac{3 k T}{m} ), the pressure becomes
      P=nkT=NkTVP = n k T = \frac{N k T}{V}
    • In molar form (using molar density $nm = NA n = \dfrac{N}{V}$ and $R = N_A k$):
      PV=NkT=nRT⇒PV=nRTP V = N k T = n R T \quad \Rightarrow \quad P V = n R T
  • Notes on the derivation style and assumptions
    • This sketch uses a kinetic-theory framework with classical, non-interacting molecules in thermal equilibrium.
    • Assumes elastic collisions with the container walls and isotropy of velocity components.
    • Three-dimensional treatment relies on the fact that the velocity components are equally distributed on average: ⟨vx^2⟩ = ⟨vy^2⟩ = ⟨v_z^2⟩ = ⟨v^2⟩/3.
    • The result connects microscopic momentum transfer to the macroscopic gas law: P V = N k T or P V = n R T.

Practical takeaways and exam-ready points

  • The Maxwell–Boltzmann distribution describes the probability density of speeds in a classical ideal gas at temperature $T$.
  • Three characteristic speeds are commonly cited:
    • Most probable speed: vmp=2kTmv_{\text{mp}} = \sqrt{\dfrac{2 k T}{m}}
    • Mean speed: ⟨v⟩=8kTπm\langle v \rangle = \sqrt{\dfrac{8 k T}{\pi m}}
    • RMS speed: vrms=3kTmv_{\text{rms}} = \sqrt{\dfrac{3 k T}{m}}
  • Mass and temperature control the distribution shape:
    • Higher $m$ shifts the distribution to lower speeds (heavier molecules move more slowly on average).
    • Higher $T$ broadens the distribution and shifts it toward higher speeds.
  • Expectations for the distribution shape and limits:
    • $f(v) \to 0$ as $v \to 0^+$ and $f(v) \to 0$ as $v \to \infty$; the peak lies at $v_{\text{mp}}$.
    • The distribution is skewed to the right, so the mean speed is greater than the most probable speed.
  • Connection to real-world measurements and theory:
    • Matches observations of molecular speeds in gases at moderate temperatures where classical statistics apply.
    • Provides a bridge from microscopic kinetics to macroscopic thermodynamic quantities like pressure and temperature.
  • Contextual links:
    • Builds on the ideal gas law and the kinetic theory of gases.
    • Uses the Boltzmann distribution for kinetic energy and the relationship between microscopic energy and speed.
    • Sets the stage for understanding gas transport, diffusion, viscosity, and heat capacity from a molecular perspective.
  • Reading and assessment reminders:
    • The reading assignments themselves are not graded; focus is on understanding the derivations and concepts.
    • Be prepared to comment on Canvas with the specific textbook sections you read when assignments are posted.

Closing logistical note

  • The lecturer signaled a break and noted the plan to continue with the pressure derivation and related concepts in the next class (Wednesday).
  • If you want to study efficiently, sketch the Maxwell distribution for varying $m$ and $T$, and derive the three speed relationships from the distribution integrals.