Heat capacity and specific heat capacity

The energy needed for a temperature rise depends on what is being heated and how much material there is.

Heating transfers energy into a sample's internal store. This page considers a temperature change without a change of state, with specific heat capacity treated as constant over the interval.

A property of the body and a property per unit mass

Heat capacity, C

The energy needed to raise the temperature of the specified body by one degree Celsius or one kelvin. Its unit is J/°C or J/K. A larger amount of the same material has a larger heat capacity under the same conditions.

Specific heat capacity, c

The energy needed to raise the temperature of unit mass of a substance by one degree Celsius or one kelvin. A common unit is J/(kg °C), equivalent to J/(kg K). The word specific means that the energy is given per unit mass.

Here ΔT denotes a temperature difference: final temperature minus initial temperature on one scale. It can be expressed in °C or K, matching the heat-capacity unit. A change of 1°C is the same size as a change of 1 K. This does not mean that a temperature of 1°C equals 1 K.

Match the mass unit to the supplied value

A value of 4200 J/(kg °C) means that 1 kg requires 4200 J for a 1°C rise. The equivalent value is 4.2 J/(g °C), because 1 kg contains 1000 g. This is also 4.2 J/(g K), since a temperature interval of 1 K equals an interval of 1°C.

For 100 g of that material warming by 10°C, either use 0.100 x 4200 x 10 or use 100 x 4.2 x 10. Both give 4200 J. Using 100 with the kilogram value would mix incompatible units.

Separate energy reaching the sample from heater input

Power is energy transferred per unit time. If a 120 W heater transfers all its output to the metal sample above at a constant rate, the ideal time is t = E/P = 2700/120 = 22.5 s.

If some heater energy warms the apparatus or transfers to the surroundings, less reaches the sample in that time. Reaching the same 30°C rise then requires more input energy or a longer time. A heater's output is not automatically the sample's energy gain.

Determine a specific heat capacity

For a suitable metal sample, use an embedded heater of known constant power and a temperature sensor in good thermal contact with the sample. Insulation reduces transfer to the surroundings. Keep the sample below any change of state.

1. Measure the mass: use a balance for the sample itself, excluding separate supports and instruments.
2. Measure the temperature rise: record the initial and final sample temperatures over the chosen heating interval. Place the sensor in the sample's intended sensing position, rather than measuring the exposed heater. Choose suitable range and resolution, and allow for response lag.
3. Find the supplied energy: with constant heater power P and measured time t, E_input = Pt.
4. Use the energy account: the input warms the sample and apparatus and may escape to the surroundings. If these other transfers are negligible or have been estimated and subtracted, c = E_sample / (mΔT).

If all input is wrongly treated as energy reaching the sample, the numerator is too large and the calculated c tends to be too high. Better insulation and accounting for apparatus heating address that cause. Repeating the same imperfect setup may show variation but does not remove the systematic error.

For cooling, say which energy you are calculating

If the same 0.20 kg sample cools from 50°C to 20°C without changing state, the positive amount of energy transferred out is mc(50 - 20) = 2700 J. Under the stated model, the sample's internal-energy change is -2700 J.

Choose the temperature interval and the system. E = mcΔT does not cover melting or boiling, and Pt is the heater input unless the question says all of it reaches the sample.