Specific Heat

Formula

Heat absorbed (or released) when a substance changes temperature by ΔT:

Q = m · c · ΔT

where:

• Q = heat (J).
• m = mass (kg).
• c = specific heat capacity (J/(kg·K)).
• ΔT = temperature change (K or °C, same numerically).

For molar form (per mole):

Q = n · C · ΔT

where C = c × molar_mass.

Specific heats of common substances

Substance	c (J/(kg·K))	Notes
Hydrogen gas (H₂)	14,300	Highest of any common substance
Helium gas	5,193	Used in lab cryogenics
Liquid water	4,186	Reference; extraordinarily high
Ice (solid water)	2,108	Half of liquid water
Steam (water gas)	2,010	—
Wood (dry)	1,700	Reasonable thermal mass
Air (dry)	1,005	—
Aluminum	897	Cookware, heat sinks
Glass	840	—
Granite	790	Thermal mass in passive solar
Iron	449	—
Copper	385	Used in radiators
Silver	235	Excellent conductor, low thermal mass
Gold	129	—
Lead	129	—
Mercury (liquid)	140	1/30 of water!

C_p vs C_v

For gases, two specific heats matter:

Symbol	Process	For ideal gas
C_p	Constant pressure	(f+2)/2 · R
C_v	Constant volume	f/2 · R
C_p - C_v	Difference	R (always for ideal gas)
γ = C_p/C_v	Adiabatic index	(f+2)/f

where f = degrees of freedom (3 for monoatomic, 5 for diatomic at room T, 6+ for polyatomic at high T).

Gas	f	C_v (J/(mol·K))	C_p (J/(mol·K))	γ
Monoatomic (He, Ar)	3	12.47	20.79	1.667
Diatomic (N₂, O₂)	5	20.79	29.10	1.400
Polyatomic (CO₂)	6	~25	~33	~1.30

Water's role in climate

Water's high c (4186 J/(kg·K)) makes it Earth's primary heat reservoir. Compare:

• Atmosphere — total mass ~5 × 10¹⁸ kg, c = 1005. Total heat capacity ~5 × 10²¹ J/K.
• Oceans — total mass ~1.4 × 10²¹ kg, c = 4186. Total heat capacity ~6 × 10²⁴ J/K.
• Ratio: Ocean : Atmosphere ≈ 1000 : 1.

Oceans absorb ~93% of warming caused by greenhouse gases. They release heat slowly, smoothing out seasonal and decadal variations. Coastal climates are mild; continental are extreme.

JavaScript — specific heat calculations

// Heat needed for temperature change
function heatRequired(mass, c, deltaT) {
  return mass * c * deltaT;
}

// Heat 1 kg of water from 25°C to 100°C
const c_water = 4186;
console.log(`Boil 1 kg water from 25°C: ${heatRequired(1, c_water, 75).toFixed(0)} J`); // 313,950

// Compare to heating same mass of iron
const c_iron = 449;
console.log(`Heat 1 kg iron 25°C → 100°C: ${heatRequired(1, c_iron, 75).toFixed(0)} J`); // 33,675

// Same heat input, different ΔT
function tempChange(mass, c, Q) {
  return Q / (mass * c);
}

const Q = 100000;  // 100 kJ
console.log(`100 kJ into 1 kg water: ΔT = ${tempChange(1, c_water, Q).toFixed(1)}°C`); // ~24
console.log(`100 kJ into 1 kg iron: ΔT = ${tempChange(1, c_iron, Q).toFixed(1)}°C`);   // ~223

// Energy to heat a swimming pool
function poolEnergy(volume_m3, deltaT, density = 1000, c = c_water) {
  const mass = volume_m3 * density;
  return heatRequired(mass, c, deltaT);
}

const pool = poolEnergy(2500, 15);  // Olympic pool, 15°C rise
console.log(`Heat Olympic pool 15°C: ${(pool / 1e9).toFixed(1)} GJ = ${(pool / 3.6e6).toFixed(0)} kWh`);

// Mixing problem: hot tea + cold milk → temperature?
function mixedTemp(masses, temps, capacities) {
  // Conservation: Q lost by hot = Q gained by cold
  // m_1·c_1·(T - T_1) = -m_2·c_2·(T - T_2) ... = 0
  // T·Σ(m·c) = Σ(m·c·T)
  let num = 0, den = 0;
  for (let i = 0; i < masses.length; i++) {
    num += masses[i] * capacities[i] * temps[i];
    den += masses[i] * capacities[i];
  }
  return num / den;
}

// 200 g tea at 90°C + 50 g milk at 5°C
console.log(mixedTemp([0.2, 0.05], [90, 5], [c_water, c_water]).toFixed(1));  // ~73°C

// Adiabatic index for ideal gas
function adiabaticIndex(degrees_of_freedom) {
  return (degrees_of_freedom + 2) / degrees_of_freedom;
}

console.log(`Monoatomic: γ = ${adiabaticIndex(3).toFixed(3)}`);  // 1.667
console.log(`Diatomic: γ = ${adiabaticIndex(5).toFixed(3)}`);    // 1.4

Where specific heat shows up

• Cooking. Why water boils, how long to simmer, oven preheating times. Sauces with butter heat differently from broth.
• Climate science. Ocean thermal capacity dominates Earth's heat budget; ocean heat content is key climate metric.
• Building design. Thermal mass (concrete, water, brick) for passive solar; insulation balances against thermal mass for comfort.
• Engineering. Coolant choice (water for high heat capacity; oil for higher temps; ammonia for refrigeration), heat exchanger design.
• Material science. Differential scanning calorimetry measures c vs T to study phase transitions and material properties.
• Geothermal. Earth's heat capacity allows seasonal and daily heat storage in shallow ground.
• Cryogenics. Liquid helium (low c, low T) for cooling MRI magnets and quantum computers.

Common mistakes

• Confusing specific heat with heat capacity. Specific heat c is per unit mass. Total heat capacity C = m·c.
• Using wrong c for state. Liquid water (4186), ice (2108), steam (2010) — all different. Phase changes have separate latent heats.
• Ignoring temperature dependence. Specific heats vary with T (small for liquids, more for solids and gases at high T). For most everyday situations, treat as constant.
• Mixing C_p and C_v for gases. Constant pressure vs constant volume gives different c values for gases. Use the right one for your process.
• Misapplying mass. Q = m·c·ΔT — make sure m is the actual mass that's being heated. For a partly-filled container, only the contents (and container if heated) count.
• Treating phase changes with c. Specific heat doesn't apply during phase changes (T doesn't change). Use latent heat L instead — Q = m·L.