Chapter 13 — Kinetic Theory

Fig 1 — Educational Diagram: Introduction and Basic Assumptions

What is Kinetic Theory?
Kinetic theory of gases gives a microscopic explanation of the macroscopic properties of gases (pressure, temperature, etc.) by modelling a gas as a large number of small particles (molecules) in constant, random motion. It connects molecular motion with thermodynamic quantities and explains gas laws (e.g., ideal gas law).

Purpose: To explain pressure, temperature and other properties of an ideal gas in terms of molecular motion and statistical averages.

Basic assumptions (ideal gas model)

1. Number of molecules is very large: The gas contains a very large number of identical molecules (so statistical treatment is meaningful).
2. Point-like molecules: The molecules are treated as point particles whose own volume is negligible compared to the volume of the container.
3. No long-range intermolecular forces: Except during collisions, molecules exert no forces on each other (no attraction or repulsion).
4. Elastic collisions: Collisions between molecules and between molecules and the walls are perfectly elastic (total kinetic energy conserved in collisions).
5. Random motion: Molecules move in random directions with a distribution of speeds and obey Newton's laws between collisions.
6. Time between collisions large compared to collision duration: Motion is free (straight-line) between brief collisions.
7. Identical masses: All molecules (for a pure gas) have the same mass m.

Consequences / Physical ideas

• Pressure on walls: Pressure arises from innumerable collisions of molecules with container walls; average momentum transfer per collision times collision rate gives pressure.
• Temperature and kinetic energy: Absolute temperature T is a measure of average translational kinetic energy of molecules. Higher T → higher average kinetic energy → higher molecular speeds.
• Ideal gas law from microscopic view: Using the assumptions and averaging over directions one obtains pV = (1/3) Nm <v^2> and, with the thermodynamic relation, pV = Nk_B T.

When assumptions fail: At high pressure or low temperature molecular sizes and intermolecular forces become important (real gases). Then ideal-gas assumptions break down and corrections (e.g., van der Waals equation) are needed. Also gases with internal degrees of freedom (rotational, vibrational) store energy not only in translational motion.

Notation used: N = total number of molecules, V = volume, n = number of moles, m = mass of one molecule, M = molar mass (kg mol^-1), k_B = Boltzmann constant, R = universal gas constant, <v^2> = mean square speed, v_rms = sqrt(<v^2>).

Fig 2 — Educational Diagram: Equation of State for Ideal Gases

What it states (macroscopic form)
The equation of state for an ideal gas relates pressure (P), volume (V), temperature (T) and the amount of gas (n): PV = nRT. Here R is the universal gas constant (R = 8.314 J·mol^-1·K^-1). For N molecules the form is PV = NkT, where k is the Boltzmann constant (k = 1.38×10^-23 J·K^-1) and N = nN_A.

Derivation from kinetic theory (brief)
Consider N identical molecules of mass m in a container of volume V. For motion in three dimensions, using the average of the squared velocity components, kinetic theory yields the pressure produced by molecular collisions as:
P = (1/3) (N/V) m <v^2>, where <v^2> is the mean square speed (average of v² over all molecules). The microscopic definition of temperature gives the relation between kinetic energy and temperature:
(1/2) m <v^2> = (3/2) kT. Substituting <v^2> into the pressure expression gives PV = NkT, which for N = nN_A becomes PV = nRT.

Physical meaning and consequences
- Temperature is a measure of the average translational kinetic energy of molecules: average kinetic energy per molecule = (3/2)kT.
- Internal energy of a monatomic ideal gas (only translational degrees of freedom) is U = (3/2) nRT (depends only on T).
- From the equation of state follow the gas laws: Boyle’s law (PV = constant at fixed T), Charles’s law (V ∝ T at fixed P), Gay-Lussac’s law (P ∝ T at fixed V).

Assumptions behind the ideal gas equation
1) Molecules are point particles (negligible volume). 2) No intermolecular forces except elastic collisions. 3) Collisions with walls and between molecules are perfectly elastic. 4) Large number of particles and classical (non-quantum) behavior. These assumptions hold well at low pressure and high temperature; deviations occur for real gases near high pressure/low temperature (condensation).

Corrections for real gases
Real gases are described by corrected equations like Van der Waals: (P + a(n/V)^2)(V - nb) = nRT, where a and b account for intermolecular attractions and finite molecular size.

Fig 3 — Educational Diagram: Molecular Explanation of Pressure

What is pressure at molecular level?

Macroscopic pressure of a gas on the walls of its container arises from countless microscopic impacts of moving gas molecules. Each time a molecule collides elastically with the wall it transfers momentum to the wall. The average rate of momentum transfer per unit area is the pressure.

Derivation (cube of side L)

1. Consider N identical molecules (mass m) enclosed in a cube of side L (volume V = L³). Take the x-axis perpendicular to one wall.
2. For a molecule with x-component of velocity v_x, an elastic collision with the wall reverses this component, so the change in momentum is Δp_x = 2 m v_x.
3. The time between successive collisions of this molecule with the same wall is Δt = 2L / |v_x|. So average force exerted by this molecule on that wall is F = Δp_x / Δt = (2 m v_x) / (2L / |v_x|) = m v_x^2 / L.
4. Total force on the wall from all N molecules is F_total = (m / L) Σ v_{x,i}^2 (sum over all molecules).
5. Pressure p = F_total / Area = F_total / L^2 = (m / V) Σ v_{x,i}^2.
6. Assuming isotropy of molecular motion, average kinetic energy is equally shared among components: Σ v_{x,i}^2 = (1/3) Σ v_i^2. Therefore

Key result:

p = (1/3) (N m / V) <v^2> = (1/3) ρ <v^2>, where <v^2> is the mean square speed and ρ = mass density = N m / V.

Relation to temperature and kinetic energy

Define root-mean-square speed v_rms = <v^2>^{1/2}. The average translational kinetic energy per molecule is (1/2) m <v^2>. Statistical mechanics/kinetic theory gives

(1/2) m <v^2> = (3/2) k_B T,

so combining with p = (1/3) (N m / V) <v^2> leads to the ideal gas law

pV = N k_B T = n R T.

Assumptions used

• Gas consists of a large number of identical point-like molecules.
• Molecules move randomly and isotropically; no preferred direction.
• Collisions with the walls and between molecules are perfectly elastic.
• Intermolecular forces are negligible except during collisions (ideal gas approximation).

Physical picture

Higher molecular speeds or greater molecular density (more molecules per unit volume) increase the average momentum transfer per unit time and thus increase pressure. Heating a gas raises molecular speeds and therefore pressure if volume is held fixed.

Fig 4 — Educational Diagram: Kinetic Interpretation of Temperature

Basic idea: Temperature is a measure of the average random kinetic energy of the microscopic particles (atoms or molecules) in a substance. In kinetic theory of gases, macroscopic temperature is directly related to the translational motion of gas particles: higher temperature means higher average random speeds.

Derivation (ideal monoatomic gas):

• From kinetic theory (by considering momentum transfer on walls) pressure p of a gas of N molecules (each of mass m) in volume V is given by: pV = (1/3) N m <v^2>, where <v^2> is the mean square speed.
• From thermodynamics, for an ideal gas pV = NkT, where k is Boltzmann constant and T is absolute temperature.
• Equating the two expressions: NkT = (1/3) N m <v^2> → kT = (1/3) m <v^2>.
• Rewriting gives the kinetic interpretation: (1/2) m <v^2> = (3/2) kT. Thus the average translational kinetic energy per particle = (3/2) kT.

Consequences and remarks:

• Total translational kinetic energy of N particles: K_total = (3/2) N k T. Per mole: K_total = (3/2) n R T (n = number of moles).
• RMS speed: v_rms = sqrt(<v^2>) = sqrt(3kT/m) = sqrt(3RT/M), where M is molar mass.
• Equipartition theorem: each independent quadratic degree of freedom contributes (1/2) kT to the average energy per particle. For a monatomic gas only 3 translational degrees exist, giving (3/2) kT. Diatomic and polyatomic gases have additional rotational (and at higher T, vibrational) contributions.
• Temperature refers to random microscopic motion only; bulk macroscopic motion (e.g., whole gas flowing) does not raise temperature unless randomized (thermalized).
• Absolute zero (T = 0 K) corresponds, in the classical picture, to zero average kinetic energy. Quantum mechanics modifies the behavior at very low temperatures (zero-point energy, frozen degrees of freedom).
• Limitations: relations above assume ideal gas behavior, classical equipartition (valid when kT is large compared to energy level spacing). At low T or for strongly interacting/condensed systems, quantum/statistical corrections are needed.

How a thermometer works (brief): A thermometer equilibrates thermally with its surroundings. Energy exchange changes microscopic kinetic energies inside the thermometer material (e.g., liquid expands, resistance changes), and that macroscopic change is calibrated to give temperature.

Fig 5 — Educational Diagram: Velocity and Speed Distributions

Overview: In the kinetic theory of gases, the motion of molecules in a gas is described statistically. Each molecule has a velocity vector (with components v_x, v_y, v_z) and a speed v = |v⃗|. The distributions tell us the probability of finding molecules with particular velocity components or speeds.

Velocity components (one-dimensional): The distribution of any single velocity component (e.g. v_x) is a Gaussian (normal) centered at 0 because motion is equally probable in ± directions:

f(v_x) = √[m/(2πkT)] · exp[−m v_x²/(2kT)]

This function is symmetric about v_x = 0 and normalized so ∫_−∞^∞ f(v_x) dv_x = 1. The width (variance) increases with temperature T and decreases with particle mass m.

Speed distribution (Maxwell–Boltzmann): Speed is the magnitude of velocity; its distribution in 3D is the Maxwell speed distribution:

f(v) = 4π · v² · (m / 2πkT)^3/2 · exp[−m v²/(2kT)],   v ≥ 0

Here f(v) dv is the probability that a molecule has speed between v and v + dv. The factor v² arises from the volume element in 3D velocity space (more states at higher v) and the exp(−mv²/2kT) factor gives the Boltzmann weight.

Characteristic speeds:

• Most probable speed (peak of f): v_mp = sqrt(2kT / m)
• Average (mean) speed: <v> = sqrt(8kT / πm)
• Root-mean-square speed: v_rms = sqrt(3kT / m)

These satisfy v_mp < <v> < v_rms. The mean kinetic energy per molecule is (1/2)m v_rms² = (3/2) kT.

Dependence on T and m: Increasing temperature broadens the distributions and shifts them to higher speeds; heavier particles (larger m) have narrower distributions and lower characteristic speeds at the same T.

Normalization and moments (quick): ∫₀^∞ f(v) dv = 1. Moments (e.g. <vⁿ>) can be obtained by integrating vⁿ f(v) dv; the common ones above give v_mp, <v>, v_rms.

Physical meaning: The distributions quantify how speeds are shared among particles. Even in a single gas sample at fixed T, molecules have a wide range of speeds — some much faster than the average (long tail of distribution).

Fig 6 — Educational Diagram: Characteristic Speeds

What are characteristic speeds? For a gas described by the Maxwell–Boltzmann velocity distribution, three characteristic speeds are commonly used to summarise the distribution of molecular speeds: the most probable speed (v_mp), the mean (average) speed (v_avg), and the root-mean-square speed (v_rms). They give different measures of a typical molecular speed and are useful in kinetic theory calculations.

Maxwell–Boltzmann speed distribution (3D):

f(v) = 4π (m / 2πkT)^{3/2} v^2 e^{−mv^2/(2kT)}, where v is speed, m is mass of one molecule, k is Boltzmann constant and T is absolute temperature.

Definitions and brief derivations (outline):

• Most probable speed v_mp: the speed where f(v) is maximum. Set df/dv = 0 → v_mp = √(2kT/m).
• Mean (average) speed v_avg: v_avg = ∫_0^∞ v f(v) dv = √(8kT / πm).
• Root-mean-square speed v_rms: v_rms = √(⟨v^2⟩) with ⟨v^2⟩ = ∫_0^∞ v^2 f(v) dv = √(3kT/m).

Using molar quantities (m = M / N_A and k = R / N_A) these can be written with the molar mass M (in kg mol^{-1}) and the gas constant R:

• v_mp = √(2RT / M)
• v_avg = √(8RT / πM)
• v_rms = √(3RT / M)

Order and numeric ratios: v_mp < v_avg < v_rms. Their ratios (relative to v_mp) are: v_avg / v_mp = 2 / √π ≈ 1.128, and v_rms / v_mp = √(3/2) ≈ 1.225. So the three speeds are close but distinct.

Physical meaning and relations:

• v_mp gives the most likely speed of a molecule in the ensemble.
• v_avg is the arithmetic mean of speeds — relevant for processes depending linearly on speed (e.g., mean collision frequency per molecule ∝ v_avg).
• v_rms is directly related to kinetic energy: (1/2) m v_rms^2 = (3/2) kT, so v_rms is useful when connecting macroscopic temperature to microscopic kinetic energy.

Assumptions and applicability: These formulas assume an ideal classical gas obeying Maxwell–Boltzmann statistics (low density, not quantum-degenerate). At very low temperatures or very high densities quantum effects modify the distribution.

Quick numeric example: For nitrogen (M ≈ 0.028 kg mol^{-1}) at T = 300 K: v_mp ≈ √(2RT/M) ≈ 457 m s^{-1}, v_avg ≈ 511 m s^{-1}, v_rms ≈ 517 m s^{-1} (values rounded).

Fig 7 — Educational Diagram: Degrees of Freedom and Equipartition Theorem

Degrees of Freedom (DoF) — Definition: Degrees of freedom of a molecule are the independent coordinates in which its atoms can store energy (kinetic or potential). For a system of N atoms, total available mechanical degrees = 3N (three coordinates per atom).

Classification of DoF:

• Translational: motion of the whole molecule in x, y, z (3 DoF for any molecule).
• Rotational: rotation about axes through centre of mass. Linear molecules: 2 rotational DoF; Nonlinear molecules: 3 rotational DoF.
• Vibrational: internal vibrations of atoms. Number of vibrational modes = 3N − 5 for linear molecules and 3N − 6 for nonlinear molecules. Each vibrational mode has both kinetic and potential parts (counts as 2 quadratic degrees when classically active).

Examples of counting: Monoatomic gas (He, Ar): N=1, total DoF = 3 (all translational). Diatomic rigid molecule (O₂, N₂) at moderate T: translational 3 + rotational 2 = 5 active DoF; vibrational modes often frozen at low T. Nonlinear triatomic (H₂O): 3N=9 so 3 translational + 3 rotational + (9 − 6) = 3 vibrational modes.

Equipartition Theorem — Statement: In classical statistical mechanics, each independent quadratic term in the energy (Hamiltonian) contributes an average energy of (1/2)kT per molecule (or (1/2)RT per mole) in thermal equilibrium at temperature T. Here k is the Boltzmann constant and R is the gas constant.

Consequences for Ideal Gases: If a molecule has f active quadratic degrees of freedom, average energy per molecule = (f/2) kT, and per mole = (f/2) RT. For n moles, internal energy U = n (f/2) RT.

Heat capacities: Molar heat capacity at constant volume: C_V,m = (dU/dT)_V = (f/2) R. Molar heat capacity at constant pressure: C_P,m = C_V,m + R = (f+2)/2 R.

Typical classical predictions: - Monoatomic gas (f=3): C_V,m = 3/2 R. - Diatomic gas (rigid, ignoring vibrations, f=5): C_V,m = 5/2 R. - Diatomic at very high T when vibration active (f=7): C_V,m = 7/2 R.

Limitations and Quantum Corrections: Equipartition is classical. Quantum mechanics shows that a mode is excited only if kT is comparable to or larger than the energy spacing of that mode (e.g. vibrational quantum hν). At low T many vibrational (and sometimes rotational) modes are "frozen out," so measured heat capacities are lower than classical values. This explains temperature dependence of specific heats and why Dulong–Petit (classical) fails at low T for solids.

Relation to kinetic theory and pressure: For monoatomic ideal gas, average translational kinetic energy per molecule = (3/2) kT and PV = (2/3) N <KE> = NkT, recovering ideal gas law PV = NkT = nRT.

Summary: Degrees of freedom tell how energy can be partitioned among motions. Equipartition assigns (1/2)kT energy to each quadratic degree at equilibrium, giving simple formulae for internal energy and heat capacities, but quantum effects can restrict applicability at low temperatures or for high-frequency modes.

Fig 8 — Educational Diagram: Internal Energy of Ideal Gases

Definition: Internal energy (U) of an ideal gas is the total microscopic energy of its molecules. For an ideal gas (no intermolecular potential), this energy is entirely kinetic: translational, and where applicable, rotational and vibrational energies of the molecules.

Key idea: For ideal gases the internal energy depends only on temperature, not on pressure or volume. This follows from the kinetic theory and the equipartition theorem.

Monoatomic ideal gas (e.g., He, Ne):
Each molecule has only 3 translational degrees of freedom. From kinetic theory the average kinetic energy per molecule is (3/2)kT, where k is Boltzmann's constant. For N molecules,
U = N × (3/2) kT = (3/2) NkT. Converting to moles (N = nN_A and kN_A = R) gives
U = (3/2) nRT.

General case — degrees of freedom (f):
By the equipartition theorem, each quadratic degree of freedom contributes (1/2)kT per molecule. If a molecule has f active degrees of freedom,
average energy per molecule = (f/2) kT,
total internal energy = U = N (f/2) kT = n (f/2) RT. Examples: f = 3 for monoatomic, f = 5 for diatomic (translational + two rotational) at ordinary temperatures, f increases when vibrational modes are excited at higher T.

Relation with heat capacity:
For ideal gases, U is a function of temperature only, so ΔU = n C_v ΔT, where C_v is molar specific heat at constant volume. From equipartition,
C_v = (f/2) R.

Implications & limitations:
- Internal energy change depends only on temperature: if temperature is constant, ΔU = 0 even if P or V change.
- Real gases deviate from ideal behaviour at high pressure / low temperature because of intermolecular potential energy, so internal energy then includes potential contributions and depends on volume too.
- At low temperatures some degrees of freedom (especially vibrational) may be ‘‘frozen out’’ quantum mechanically, so classical equipartition predictions fail.

Short derivation (monoatomic):
Average translational kinetic energy per molecule = (1/2) m v_rms^2 = (3/2) kT. Multiply by N molecules: U = (3/2) NkT = (3/2) nRT.

Fig 9 — Educational Diagram: Specific Heats and Ratio of Specific Heats

Specific heat is the heat required to raise the temperature of a unit amount (mass or amount of substance) of a substance by 1 K. For gases we commonly use:

• Cv (specific heat at constant volume): heat required per mole (or per kg) to raise temperature by 1 K at constant volume.
• Cp (specific heat at constant pressure): heat required per mole (or per kg) to raise temperature by 1 K at constant pressure.

For an ideal gas, at constant volume all heat supplied increases internal energy U, so dQv = dU and Cv = (1/n)(dU/dT) (molar basis). At constant pressure dQp = dH where H is enthalpy, so Cp = (1/n)(dH/dT).

Kinetic theory / Equipartition theorem: each quadratic degree of freedom (d.o.f.) of a molecule contributes (1/2)kT per molecule to its average energy (or (1/2)RT per mole). If a molecule has f active quadratic degrees of freedom, its molar internal energy is U_m = (f/2) RT. Hence the molar specific heat at constant volume is Cv,m = (dU_m/dT) = (f/2) R. Using H = U + pV = U + RT (per mole), Cp,m = Cv,m + R = (f+2)/2 * R.

Ratio of specific heats (gamma) is defined as γ = Cp/Cv. For an ideal gas γ = Cp,m / Cv,m = (f+2)/f. γ is important in adiabatic processes and acoustics.

Internal energy and enthalpy (molar): U_m = (f/2) RT, H_m = U_m + RT = ((f+2)/2) RT. For n moles: U = n (f/2) RT.

Useful consequences (ideal gas): - Cp,m − Cv,m = R - For adiabatic (reversible, ideal gas): PV^γ = constant, TV^{γ−1} = constant, and T^γ P^{1−γ} = constant. - Speed of sound in an ideal gas: v = sqrt(γ R T / M), where M is molar mass.

Physical meaning & temperature dependence: f is the number of active degrees of freedom. Typical values: - Monatomic gases (He, Ne, Ar): f = 3 → Cv,m = 3/2 R, Cp,m = 5/2 R, γ = 5/3 ≈ 1.67. - Diatomic gases at room temperature (N2, O2): translational + rotational active → f = 5 → Cv,m = 5/2 R, Cp,m = 7/2 R, γ = 7/5 = 1.4. - At higher temperatures vibrational modes activate increasing f and changing Cv and γ. Real gases deviate from these ideal values and specific heats vary with temperature.

Practical notes: Use molar values (Cv,m, Cp,m) when working with moles; use mass-specific values (c_v, c_p) when working with kilograms: c_v = Cv,m / M and c_p = Cp,m / M, where M is molar mass.

Fig 10 — Educational Diagram: Mean Free Path and Collision Frequency

Definition: The mean free path (λ) of a gas molecule is the average distance it travels between successive collisions with other molecules. The collision frequency (z) is the average number of collisions a molecule suffers per unit time.

Assumptions (Kinetic theory): gas is dilute; molecules are hard spheres of diameter d; only binary collisions matter; distribution of molecular speeds is Maxwellian; intermolecular forces are negligible except during collisions.

Derivation (sketch): Consider one molecule moving in a sea of other molecules with number density n (number per unit volume). In time in which it travels a distance λ it sweeps a cylinder of volume σ·λ, where σ is the effective collision cross-section. For identical hard spheres σ = πd². The average number of target centers in this volume is n·σ·λ. To account for the fact that all molecules move (so relative motion increases collision rate), a factor √2 appears when using statistical relative speeds. Setting the average number of collisions in the distance λ equal to 1 gives:

λ = 1 / (√2 · n · σ).

Using the ideal-gas relation n = p/(k_B·T) (k_B = Boltzmann constant) we get a useful engineering form:

λ = k_B·T / (√2 · π · d² · p).

Collision frequency: The mean speed of molecules (average of speed, not rms) is v̄ = √(8k_B·T / πm). The average collision frequency per molecule (collisions per second) is

z = v̄ / λ = √2 · n · σ · v̄.

Equivalently, the mean time between collisions (mean free time) is τ = 1 / z = λ / v̄.

Collision rate per unit volume: The number of collisions occurring per unit volume per unit time (counting every collision once) is

R = (1/2) · n² · σ · <v_rel> = (1/2) · n² · σ · √2 · v̄ = (n² · σ · v̄) / √2.

Dependence on p and T: At constant temperature, λ ∝ 1/p (inverse of pressure). At constant pressure, λ ∝ T; v̄ ∝ √T, so z = v̄/λ ∝ 1/√T (decreases slowly with increasing T if p fixed). If number density n is fixed (e.g., sealed container), λ is independent of T and z ∝ √T.

Typical scale: For air at STP (T ≈ 300 K, p ≈ 1 atm) and molecular diameter of order 3×10⁻¹⁰ m, λ is of order 10⁻⁷ m (tens of nanometres). In high vacuum λ can grow to centimeters, meters or more depending on pressure.

Physical significance: Mean free path separates microscopic (collision-dominated) and macroscopic (continuum) behavior. When characteristic macroscopic length L ≫ λ, continuum (hydrodynamic) assumptions hold; when L ~ λ or smaller, free-molecular or rarified-gas effects dominate (important in vacuum technology, spacecraft upper-atmosphere drag, micro/nano flow).

Fig 11 — Educational Diagram: Deviations from Ideal Behaviour and Limitations

Overview

Ideal-gas behaviour (PV = nRT) is based on simplified assumptions of the kinetic theory: point-like molecules, no intermolecular forces, perfectly elastic collisions, large number of molecules, and classical motion. Real gases deviate from ideal behaviour when these assumptions are not valid — typically at high pressures and low temperatures. Two main physical causes of deviation are:

• Finite molecular size (excluded volume): molecules occupy space, so the free volume available for motion is less than the container volume.
• Intermolecular forces (attraction and repulsion): attractive forces reduce the momentum transfer to the walls (lower pressure) at moderate distances; repulsive forces dominate at very short distances (higher pressure than ideal).

Consequences

• At low pressure and/or high temperature the two effects are small and gases approximately follow ideal-gas law.
• At moderate pressures attractive forces make measured pressure smaller than ideal prediction (compressibility factor Z < 1). At very high pressures repulsive volume effects dominate and Z > 1.
• Below a certain temperature (critical region) real-gas isotherms deviate strongly and show liquefaction; ideal-gas law cannot describe phase change.

Van der Waals equation (simple correction)

Van der Waals introduced two corrections: subtract an excluded volume nb from V, and add an internal pressure correction a(n/V)^2 to P. The equation (for n moles) is

(P + a n^2 / V^2)(V - n b) = n R T

Per mole (using molar volume Vm = V/n):

(P + a / Vm^2)(Vm - b) = R T

This equation qualitatively explains liquefaction and critical behaviour (S-shaped isotherms below Tc). It also gives expressions for critical constants (from inflection point):

• Tc = 8 a / (27 b R)
• Pc = a / (27 b^2)
• Vc = 3 b

Compressibility factor

Z = PV / (n R T) (or Z = P Vm / (R T)). For an ideal gas Z = 1. For real gases Z varies with P and T:

• Z < 1 : attractive forces dominate (gas is more compressible than ideal).
• Z > 1 : repulsive/volume effects dominate (less compressible).

For van der Waals, the second virial coefficient B(T) (in virial expansion PV = RT(1 + B/Vm + ...)) is B(T) = b - a/(R T). Setting B(TB) = 0 gives the Boyle temperature TB = a / (R b) — near TB a gas behaves ideally over a wide range of pressures.

Limitations of the kinetic theory / ideal-gas model

• Does not include intermolecular forces (so cannot predict liquefaction, condensation, or attractive effects).
• Assumes point particles (ignores finite molecular volume), so fails at high densities.
• Classical treatment breaks down at very low temperatures or for very light particles — quantum effects become important.
• Ignores internal degrees of freedom (rotation, vibration) that store energy in polyatomic molecules; simple kinetic theory applies best to monatomic gases.
• Cannot handle chemical reactions, dissociation, or ionization occurring at high T or in plasmas.

Practical importance

Understanding deviations is essential for accurate design and analysis of compressors, high-pressure gas cylinders, refrigeration and liquefaction processes, and when predicting real behaviour of gases near condensation. Engineers use corrected equations of state (van der Waals, Redlich–Kwong, Soave–Redlich–Kwong, Peng–Robinson, or virial expansions) chosen for the application and range of conditions.

Fig 12 — Educational Diagram: Applications and Problem-Solving Topics

Overview: Kinetic theory connects macroscopic gas properties (P, V, T) with microscopic motion of molecules. It provides formulas for pressure, speeds (most probable, mean, rms), energy per molecule, mean free path, collision frequency, effusion/diffusion and explains specific heats via equipartition. This section emphasizes applying those results to solve typical Class 11 problems.

Key ideas and derivations (concise):

• Pressure from molecular motion: P = (1/3)(N/V)m<v^2>. Using m<v^2>/2 = (3/2)kT gives ideal gas law PV = NkT.
• Energy and temperature: Average kinetic energy per molecule = (3/2)kT. Temperature is a measure of average translational kinetic energy.
• Speed measures: Maxwell distribution of speeds f(v) gives three useful speeds: most probable v_mp, mean v_avg and root-mean-square v_rms. They satisfy v_mp < v_avg < v_rms.
• Mean free path and collisions: Mean free path λ = 1/(√2 n π d^2) = kT/(√2 π d^2 P). Collision frequency (per molecule) z = √2 n σ v̄, where σ = π d^2 and v̄ is mean speed.
• Effusion and Graham's law: Effusion rate ∝ v̄ ∝ 1/√M, so Rate1/Rate2 = √(M2/M1).
• Equipartition and specific heats: Each quadratic degree of freedom carries (1/2)kT per molecule. For f degrees of freedom, molar CV = (f/2)R and CP = CV + R. For monoatomic gas f=3 → CV,m = 3/2 R; diatomic (at ordinary T, translational+rotational) f=5 → CV,m ≈ 5/2 R.

Problem-solving strategy:

• Always convert mass units: molar mass M (g mol−1) → kg mol−1 by dividing by 1000 when using R in J mol−1 K−1.
• Use N = n·NA and k = R/NA when switching between molecule-level and mole-level formulae.
• Temperature must be in kelvin; pressure in Pa for SI. For mean free path use pressure in Pa and molecular diameter in m.
• Decide if the problem uses molecule mass m or molar mass M and pick v formulas accordingly: v_rms = √(3kT/m) = √(3RT/M).
• For distribution problems, use Maxwell f(v). For fractions above/below a speed, integrate f(v) or use standard ratios/approximations; many textbook problems ask for qualitative comparisons or use known ratios v_mp:v_mean:v_rms.
• When pressure or density changes, remember λ ∝ 1/P and collision frequency ∝ P (at fixed T).

Common question types: compute v_rms/v_mean/v_mp at given T; fraction of molecules above a speed; mean free path at given P and T; collision frequency; compare effusion rates; find heat capacities from degrees of freedom; relate macroscopic PV work to microscopic energy change.