Heat Capacity Calculator (Q = mcΔT)

This calculator answers the sensible-heat question: how much energy moves in or out of a substance when its temperature changes by a given amount.

The formula:

Q = m × c × ΔT

Where:

• Q = heat energy transferred (joules; also shown here in calories, kilocalories and BTU)
• m = mass of the substance
• c = specific heat capacity, in joules per gram per °C
• ΔT = temperature change. Positive means heat went in, negative means heat came out.

Note that ΔT is a change, not a temperature, so it is identical in °C and in kelvin. A 25 degree rise is 25 °C and 25 K. Only Fahrenheit differs, and there you divide the Fahrenheit change by 1.8 first.

Worked example, a kettle

Heating 500 g of water from 20°C to 100°C means ΔT = 80 and c = 4.186 J/g·°C.

Q = 500 × 4.186 × 80 = 167,440 J, or 167.4 kJ, or 40,019 calories, or 158.7 BTU.

A 2,000 W kettle delivers 2,000 joules every second, so this takes about 84 seconds if nothing escapes. Real kettles take a little longer, and the difference is entirely heat leaking into the room plus the energy that goes into warming the kettle itself.

Specific Heat Capacities of Common Materials

Why Water’s High Heat Capacity Matters

Water has an exceptionally high specific heat capacity compared to almost anything else, and that single fact explains a surprising amount of the world:

• Oceans moderate coastal climates, because water absorbs and releases heat slowly
• Sweating cools the body efficiently, because evaporating water carries away enormous energy
• Heating a pan of water takes far more energy than heating the metal pan itself
• Central heating systems circulate water rather than air, because a given pipe carries far more heat

Look at the bottom of the table for the contrast. Lead is 33 times easier to heat than water, gram for gram, which is why a lead sinker warms in your hand almost at once and a glass of water does not.

Calorimetry: what happens when you mix two things

The reason this formula matters more than “how much energy to heat one object” is that it lets you predict a final temperature. Put a hot object into a cold one in an insulated container and the heat lost by one equals the heat gained by the other:

m₁c₁(T_f − T₁) + m₂c₂(T_f − T₂) = 0

Solve for T_f and you get a weighted average, where the weights are mass times specific heat:

T_f = (m₁c₁T₁ + m₂c₂T₂) / (m₁c₁ + m₂c₂)

Drop a 200 g steel bolt at 300°C into 1,000 g of water at 20°C. The bolt carries 200 × 0.45 = 90 J/°C; the water carries 1,000 × 4.186 = 4,186 J/°C. So T_f = (90 × 300 + 4,186 × 20) / 4,276 = 25.9°C. The bolt is glowing hot and it barely moves the water, because the water outweighs it by a factor of 46 in heat capacity.

Calories, joules and BTU

1 calorie (cal) = 4.184 J, the heat needed to raise 1 g of water by 1°C. 1 food Calorie (kcal) = 4,184 J = 1,000 cal. 1 BTU = 1,055.06 J, the heat needed to raise 1 pound of water by 1°F.

One thing this formula cannot do

Q = mcΔT only covers temperature changes within a single phase. It says nothing about melting or boiling, where energy pours in and the temperature does not move at all. Heating ice from −10°C to steam at 110°C takes five separate steps, three of them mcΔT and two of them latent heat.

Cross a phase boundary and this calculator will understate the energy badly. Boiling that 500 g of water away needs another 1,130 kJ, nearly seven times what it took to bring it from 20°C to boiling in the first place.