Entropy

Definition

Entropy has two equivalent definitions:

1. Thermodynamic (Clausius, 1865):

dS = dQ_rev / T

For any reversible process, the change in entropy equals heat added divided by temperature.

2. Statistical (Boltzmann, 1877):

S = k_B · ln(W)

where W is the number of microscopic arrangements (microstates) consistent with the macroscopic state. k_B = 1.381 × 10⁻²³ J/K.

These give the same numerical answer; they're complementary views of the same physical quantity.

Microstates and macrostates

A macrostate is the macroscopic description (T, P, V, etc.). A microstate is the specific positions and momenta of all particles. Many microstates can correspond to the same macrostate.

For a gas of N particles in a box:

• Macrostate "all gas in left half" — relatively few microstates.
• Macrostate "uniform throughout box" — vastly more microstates.

The system spends time proportional to microstate count → mostly in high-entropy states.

Entropy of common processes

Process	ΔS	Notes
Reversible isothermal expansion of ideal gas	nR·ln(V_f/V_i)	Positive (V increases)
Isobaric heating from T_i to T_f	n·C_p·ln(T_f/T_i)	Positive if T_f > T_i
Phase change (melt or boil)	L·m / T_transition	Big positive jump
Mixing of two ideal gases	R·(n_A·ln(2) + n_B·ln(2))	Always positive
Adiabatic free expansion	nR·ln(V_f/V_i) > 0	Despite Q=0, S increases (irreversible)
Reversible cycle	0 (returns to start)	S is a state function
Heat flow from hot to cold (Q from T_h to T_c)	Q·(1/T_c - 1/T_h)	Positive if T_h > T_c (always for natural flow)

Numerical entropy values

Substance / state	Standard entropy (J/(mol·K))
Diamond (C, solid)	2.4
Graphite (C, solid)	5.7
Iron (solid)	27
Liquid water (25°C)	70
Hydrogen gas (H₂, 25°C)	131
Oxygen gas (O₂, 25°C)	205
Steam (H₂O gas, 100°C)	189
CO₂ gas	214

Solids < liquids < gases — more particle freedom → more microstates → higher S.

JavaScript — entropy calculations

const k_B = 1.380649e-23;  // J/K
const N_A = 6.02214e23;    // particles/mol
const R = 8.314;           // J/(mol·K)

// Boltzmann entropy
function boltzmannS(W) { return k_B * Math.log(W); }

// Entropy change in isothermal expansion of ideal gas
function isothermalEntropyChange(n, V_i, V_f) {
  return n * R * Math.log(V_f / V_i);
}

console.log(`1 mol gas, V doubled: ΔS = ${isothermalEntropyChange(1, 1, 2).toFixed(2)} J/K`);  // ~5.76

// Entropy change for isobaric heating
function isobaricEntropyChange(n, C_p, T_i, T_f) {
  return n * C_p * Math.log(T_f / T_i);
}

// Heat 1 mol of water from 25°C to 100°C, C_p = 75 J/(mol·K)
console.log(`Heat water 25→100°C: ${isobaricEntropyChange(1, 75, 298, 373).toFixed(2)} J/K`); // ~16.9

// Phase change entropy
function phaseChangeEntropy(latent_heat, mass, transition_T) {
  return latent_heat * mass / transition_T;
}

// Melting 1 kg of ice (L_fusion = 334,000 J/kg, T = 273.15 K)
console.log(`Melt 1 kg ice: ${phaseChangeEntropy(334000, 1, 273.15).toFixed(0)} J/K`); // ~1223

// Heat transfer entropy change (always positive for natural flow)
function heatTransferEntropy(Q, T_hot, T_cold) {
  return Q * (1/T_cold - 1/T_hot);
}

console.log(`100 J from 500K to 300K: ${heatTransferEntropy(100, 500, 300).toFixed(3)} J/K`);  // +0.133

// Free expansion (gas expands into vacuum)
function freeExpansionEntropy(n, V_i, V_f) {
  // For ideal gas, free expansion is adiabatic (Q=0) but irreversible
  // Same formula as isothermal: nR·ln(V_f/V_i)
  return n * R * Math.log(V_f / V_i);
}

// Mixing entropy (ideal mixing of two gases)
function mixingEntropy(n_A, n_B) {
  // Each gas effectively expands to total volume
  const x_A = n_A / (n_A + n_B);
  const x_B = n_B / (n_A + n_B);
  // ΔS = -R · n_total · (x_A·ln x_A + x_B·ln x_B)
  return -R * (n_A + n_B) * (x_A * Math.log(x_A) + x_B * Math.log(x_B));
}

console.log(`Mix 1 mol A + 1 mol B: ${mixingEntropy(1, 1).toFixed(2)} J/K`);  // ~5.76 J/K

Where entropy shows up

• Thermodynamics — engine efficiency. Carnot limit derived from entropy considerations.
• Chemistry — Gibbs free energy. ΔG = ΔH - T·ΔS. Reactions favorable when ΔG < 0; entropy and enthalpy contributions matter.
• Cosmology. Entropy of universe increasing → arrow of time, eventual heat death.
• Black holes. Bekenstein-Hawking entropy S = A/4ℓ_p² (area of event horizon). Connection to quantum gravity.
• Information theory. Shannon entropy H = -Σp·log p; data compression limits.
• Computing. Landauer's principle — erasing 1 bit costs k_B·T·ln 2 of energy. Foundational limit on computation efficiency.
• Statistical physics. Boltzmann's foundational work; modern field theory uses entropy throughout.

Common mistakes

• Equating entropy with "messiness." Disorder is a casual proxy. Entropy is precisely defined — microstate count or dQ_rev/T.
• Forgetting it's STATE function. S depends only on state, not path. For reversible cycle, ΔS = 0. (For irreversible process between same states, S of system can be the same, but entropy of universe still increased — heat was wasted to surroundings.)
• Misapplying 2nd law to subsystems. Entropy can decrease locally; only the TOTAL must increase. Refrigerators, life, manufacturing all do this.
• Using temperature in Celsius for dS = dQ/T. Always Kelvin — dQ at T = 0 K would be infinite ΔS, which isn't right (and 0 K can't be reached).
• Confusing entropy with energy. Energy is conserved (1st law). Entropy is NOT — it can increase. Different quantities; both are state functions but not interchangeable.
• Treating S = k·ln W as exact for any state. It's a definition for an equilibrium state with W microstates. Far-from-equilibrium systems need more careful treatment.