University Physics Study Guide: Kinetic Theory

Introduction to Kinetic Theory and Historical Context

  • Foundations of Gas Laws:

    • Boyle discovered Boyle’s Law in 1661.
    • Boyle, Newton, and others attempted to explain gas behavior by viewing gases as collections of tiny atomic particles.
    • Scientific Atomic Theory was formally established over 150 years after Boyle's initial discovery.

  • Core Premise of Kinetic Theory:

    • Explains the behavior of gases based on the idea that gas consists of rapidly moving atoms or molecules.
    • The theory assumes inter-atomic forces (short-range forces critical for solids and liquids) can be neglected in the gaseous state.
    • Primary development occurred in the 19th century by scientists including Maxwell and Boltzmann.

  • Successes of Kinetic Theory:

    • Provides a molecular interpretation of macroscopic properties like pressure and temperature.
    • Consistent with established gas laws and Avogadro’s hypothesis.
    • Correctly explains the specific heat capacities of many gases.
    • Relates measurable properties (viscosity, conduction, diffusion) to molecular parameters (size and mass).

Molecular Nature of Matter

  • Richard Feynman’s Perspective:

    • Considered the discovery that "Matter is made up of atoms" as the most significant scientific finding.
    • The Atomic Hypothesis: "All things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another."

  • Ancient Conceptions of Atoms:

    • India (Vaiseshika School): Founded by Kanada (6th century B.C.). Postulated that atoms (Paramanu) were eternal, indivisible, and infinitesimal.
      • Postulated four kinds of atoms: Bhoomi (Earth), Ap (Water), Tejas (Fire), and Vayu (Air).
      • Akasa (Space) was considered continuous and inert.
      • Combinations included dvyanuka (diatomic) and tryanuka (triatomic) molecules.
      • Size estimates in Lalitavistara (2nd century B.C.) were close to the modern order of 10−10 m10^{-10}\,m.
    • Greece: Democritus (4th century B.C.) used the word "atom" meaning "indivisible."
      • He conjectured different shapes: water atoms were smooth/round (allowing flow), earth atoms were rough/jagged (creating hardness), and fire atoms were thorny (causing burns).

  • Scientific Atomic Theory (John Dalton):

    • Proposed approximately 200 years ago to explain laws of chemical combination:
      1. Law of Definite Proportions: Compounds have a fixed proportion by mass of constituents.
      2. Law of Multiple Proportions: When elements form multiple compounds, the masses of one element combining with a fixed mass of another are in ratios of small integers.

  • Supporting Laws:

    • Gay Lussac’s Law: Gases combine chemically in volumes that are ratios of small integers.
    • Avogadro’s Law: Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.

Physical Characteristics of Molecules and States of Matter

  • Atomic Dimensions:

    • Atomic size is approximately 1 Angstrom (10−10 m10^{-10}\,m).
    • Solids: Tightly packed; atoms spaced about 2\,\text{Å} apart.
    • Liquids: Separation is similar to solids (2\,\text{Å}), but atoms are not rigidly fixed, allowing flow.
    • Gases: Interatomic distances are in tens of Angstroms. Interaction is negligible except during collisions.

  • Interatomic Forces:

    • Combine long-range attraction and short-range repulsion.
    • Atoms attract at a few Angstroms but repel when squeezed closer.

  • Mean Free Path (ll):

    • The average distance a molecule travels without colliding.
    • In gases, this is on the order of thousands of Angstroms (roughly 1000×1000 \times the size of the molecule).

  • Sub-Atomic Structure:

    • Atoms consist of a nucleus (protons and neutrons) and electrons.
    • Protons and neutrons are made of quarks.
    • The current quest includes potential string-like elementary entities.

Behaviour of Gases and the Ideal Gas Equation

  • Ideal Gas Approximation:

    • Real gases approach ideal behavior at low pressures and high temperatures (far above liquefaction/solidification points).
    • Molecular interactions are negligible in these conditions.

  • The Equation of State:

    • PV=KTPV = KT
    • For a sample, KK is proportional to the number of molecules (NN): K=NkBK = N k_B.
    • Boltzmann Constant (kBk_B): Same for all gases.
    • kB=1.38×10−23 J K−1k_B = 1.38 \times 10^{-23}\,J\,K^{-1}

  • Universal Gas Constant and Moles:

    • PV=μRTPV = \mu RT
    • μ\mu = number of moles.
    • R=NAkB=8.314 J mol−1 K−1R = N_A k_B = 8.314\,J\,mol^{-1}\,K^{-1}
    • μ=MM0=0NA\mu = \frac{M}{M_0} = \frac{0}{N_A}, where MM is mass, M0M_0 is molar mass, and NAN_A is Avogadro's number.
    • Avogadro Number (NAN_A): 6.02×1023 mol−16.02 \times 10^{23}\,mol^{-1}.
    • Molar Volume: At S.T.P. (273 K273\,K, 1 atm1\,atm), 22.4 litres22.4\,litres of gas contains NAN_A molecules.

  • Other Forms of the Ideal Gas Equation:

    • P=kBnTP = k_B n T, where nn is number density (molecules per unit volume).
    • Pρ=RTM0\frac{P}{\rho} = \frac{RT}{M_0}, where ρ\rho is mass density.

  • Specific Gas Laws:

    • Boyle’s Law: PV=constantPV = \text{constant} (at constant TT).
    • Charles’ Law: V∝TV \propto T (at constant PP).
    • Dalton’s Law of Partial Pressures: For a mixture of non-interacting ideal gases, Ptotal=P1+P2+…P_{total} = P_1 + P_2 + \dots where P1=μ1RTVP_1 = \frac{\mu_1 RT}{V}.

Kinetic Theory of an Ideal Gas: Pressure Derivation

  • Assumptions:

    1. Large number of molecules in incessant random motion.
    2. Average distance between molecules is large (10×10 \times the size of a molecule).
    3. Molecules move in straight lines between collisions.
    4. Collisions (with walls or each other) are perfectly elastic.

  • Pressure Calculation (Cube of side ll):

    • A molecule with velocity (vx,vy,vz)(v_x, v_y, v_z) hits a wall parallel to the yzyz-plane.
    • Velocity after elastic collision: (−vx,vy,vz)(-v_x, v_y, v_z).
    • Change in momentum: −mvx−(mvx)=−2mvx-mv_x - (mv_x) = -2mv_x.
    • Momentum imparted to the wall: 2mvx2mv_x.
    • Number of molecules hitting area AA in time Δt\Delta t: 12AvxΔtn\frac{1}{2} A v_x \Delta t n.
    • Total momentum transfer Q=(2mvx)(12nAvxΔt)Q = (2mv_x) (\frac{1}{2} n A v_x \Delta t).
    • P=QAΔt=nmvx2P = \frac{Q}{A \Delta t} = n m v_x^2.
    • Summing over all molecules and assuming isotropy (vx2=vy2=vz2=13v2v_x^2 = v_y^2 = v_z^2 = \frac{1}{3} v^2):
    • P=13nmvˉ2P = \frac{1}{3} n m \bar{v}^2

Kinetic Interpretation of Temperature

  • Internal Energy (EE): For an ideal gas, internal energy is purely translational kinetic energy.

    • PV=23N(12mvˉ2)=23EPV = \frac{2}{3} N (\frac{1}{2} m \bar{v}^2) = \frac{2}{3} E
    • Using PV=NkBTPV = N k_B T:
    • E=32NkBTE = \frac{3}{2} N k_B T
    • Average Kinetic Energy per Molecule: 12mvˉ2=32kBT\frac{1}{2} m \bar{v}^2 = \frac{3}{2} k_B T

  • Root Mean Square (rms) Speed (vrmsv_{rms}):

    • vrms=vˉ2=3kBTmv_{rms} = \sqrt{\bar{v}^2} = \sqrt{\frac{3 k_B T}{m}}
    • For Nitrogen at 300 K300\,K: vrms≈516 m s−1v_{rms} \approx 516\,m\,s^{-1}.
    • Lighter molecules have higher rms speeds at the same temperature.

Law of Equipartition of Energy

  • Definition: In thermal equilibrium, the total energy of a system is equally distributed among all its energy modes (degrees of freedom), with each mode contributing 12kBT\frac{1}{2} k_B T.

  • Degrees of Freedom:

    • Translational: 3 degrees (x,y,zx, y, z directions).
    • Rotational:
      • Monatomic: 0 rotational degrees.
      • Diatomic/Linear: 2 rotational degrees (axes perpendicular to the molecular bond).
    • Vibrational: Each vibrational mode contributes 2 squared terms (kinetic and potential energy), thus contributing 2×12kBT=kBT2 \times \frac{1}{2} k_B T = k_B T.

Specific Heat Capacities

  • Monatomic Gases:

    • 3 translational degrees of freedom.
    • U=32RTU = \frac{3}{2} RT
    • Cv=32R≈12.5 J mol−1 K−1C_v = \frac{3}{2} R \approx 12.5\,J\,mol^{-1}\,K^{-1}
    • Cp=52R≈20.8 J mol−1 K−1C_p = \frac{5}{2} R \approx 20.8\,J\,mol^{-1}\,K^{-1}
    • γ=CpCv=53≈1.67\gamma = \frac{C_p}{C_v} = \frac{5}{3} \approx 1.67

  • Diatomic Gases (Rigid Rotator):

    • 3 translational + 2 rotational = 5 degrees of freedom.
    • U=52RTU = \frac{5}{2} RT
    • Cv=52R;Cp=72R;γ=75=1.40C_v = \frac{5}{2} R; C_p = \frac{7}{2} R; \gamma = \frac{7}{5} = 1.40

  • Diatomic Gases (With Vibration):

    • Adds 1 vibrational mode (2 degrees).
    • U=72RTU = \frac{7}{2} RT
    • Cv=72R;Cp=92R;γ=97≈1.29C_v = \frac{7}{2} R; C_p = \frac{9}{2} R; \gamma = \frac{9}{7} \approx 1.29

  • Polyatomic Gases:

    • 3 translational, 3 rotational, and ff vibrational modes.
    • Cv=(3+f)R;Cp=(4+f)R;γ=4+f3+fC_v = (3 + f)R; C_p = (4 + f)R; \gamma = \frac{4+f}{3+f}

  • Specific Heat of Solids:

    • Each atom in a solid vibrates in 3 dimensions.
    • Each dimension has 2 degrees of freedom (KE + PE).
    • Total energy per atom = 3kBT3 k_B T.
    • For 1 mole (NAN_A atoms): U=3RTU = 3 RT.
    • C=ΔQΔT=ΔUΔT=3R≈24.9 J mol−1 K−1C = \frac{\Delta Q}{\Delta T} = \frac{\Delta U}{\Delta T} = 3 R \approx 24.9\,J\,mol^{-1}\,K^{-1}.

Mean Free Path Derivation and Estimates

  • Simplified Model:

    • Molecule diameter = dd. Average speed = ⟨v⟩\langle v \rangle.
    • A molecule sweeps a volume πd2⟨v⟩Δt\pi d^2 \langle v \rangle \Delta t in time Δt\Delta t.
    • Collisions in time Δt=nπd2⟨v⟩Δt\Delta t = n \pi d^2 \langle v \rangle \Delta t.
    • Rate of collisions = nπd2⟨v⟩n \pi d^2 \langle v \rangle.
    • Time between collisions τ=1nπd2⟨v⟩\tau = \frac{1}{n \pi d^2 \langle v \rangle}.
    • l=⟨v⟩τ=1nπd2l = \langle v \rangle \tau = \frac{1}{n \pi d^2}.

  • Refined Model (Accounting for relative motion):

    • l=12nπd2l = \frac{1}{\sqrt{2} n \pi d^2}

  • Typical Values (Air at STP):

    • n=2.7×1025 m−3n = 2.7 \times 10^{25}\,m^{-3}
    • d≈2×10−10 md \approx 2 \times 10^{-10}\,m
    • τ≈6.1×10−10 s\tau \approx 6.1 \times 10^{-10}\,s
    • l≈2.9×10−7 ml \approx 2.9 \times 10^{-7}\,m (approx 1500×1500 \times the molecular diameter).

Examples and Numerical Problems

  • Example 12.1 (Water Vapour Fraction):

    • Density of water = 1000 kg m−31000\,kg\,m^{-3}; density of vapour = 0.6 kg m−30.6\,kg\,m^{-3}.
    • Ratio of molecular volume to total volume = 0.61000=6×10−4\frac{0.6}{1000} = 6 \times 10^{-4}.

  • Example 12.2 (Water Molecule Size):

    • Molar mass of water = 18 g=0.018 kg18\,g = 0.018\,kg.
    • Mass of one molecule = 0.0186×1023=3×10−26 kg\frac{0.018}{6 \times 10^{23}} = 3 \times 10^{-26}\,kg.
    • Volume = massdensity=3×10−261000=3×10−29 m3\frac{\text{mass}}{\text{density}} = \frac{3 \times 10^{-26}}{1000} = 3 \times 10^{-29}\,m^3.
    • Radius calculation: \frac{4}{3} \pi r^3 = 3 \times 10^{-29} \Rightarrow r \approx 2\,\text{Å}.

  • Example 12.5 (Argon vs Chlorine):

    • Argon (Ar): Atomic mass 39.9 u39.9\,u. Chlorine (Cl2Cl_2): Molecular mass 70.9 u70.9\,u.
    • Average KE per molecule depends only on TT; ratio is 1:11:1.
    • vrms∝1Mv_{rms} \propto \frac{1}{\sqrt{M}}.
    • vrms(Ar)vrms(Cl)=70.939.9=1.33\frac{v_{rms}(Ar)}{v_{rms}(Cl)} = \sqrt{\frac{70.9}{39.9}} = 1.33.

  • Example 12.6 (Uranium Isotope Enrichment):

    • 235UF6{}^{235}UF_6 mass = 349349; 238UF6{}^{238}UF_6 mass = 352352.
    • Ratio of speeds = 352349=1.0044\sqrt{\frac{352}{349}} = 1.0044.
    • Percentage difference = 0.44%0.44\%.

Points to Ponder

  • Pressure Uniformity: Pressure exists throughout the fluid, not just at the walls. Internal layers are in equilibrium because pressure is equal on both sides.
  • Gravity vs. Kinetic Speed: Air molecules don't settle on the ground because their thermal kinetic energy (12mv2\frac{1}{2}m v^2) is much higher than the gravitational potential energy (mghmgh) at room heights.
  • Mean vs. Mean Square: ⟨v2⟩\langle v^2 \rangle is generally not equal to (⟨v⟩)2(\langle v \rangle)^2. The average of squares is different from the square of the average.
  • Compression and Temperature: When a gas is compressed by a piston moving inward, molecules hitting the moving piston gain speed (vrebound=2Vpiston+uv_{rebound} = 2V_{piston} + u), explaining the temperature rise during compression.