Specific Heat Capacity Equation

Linking energy transfer to temperature change

Heating transfers energy into a material's thermal energy store. If the material does not change state, this usually raises its temperature. The amount of energy required depends on three quantities:

• the mass being heated;
• the temperature change required;
• the material's specific heat capacity.

A larger mass contains more material to heat, so it requires more energy for the same temperature rise. A larger temperature rise also requires more energy. Different materials require different amounts of energy because they have different specific heat capacities.

The specific heat capacity equation

The relationship is

$$\Delta Q=m\times c\times\Delta\theta$$

where:

• $\Delta Q$ is the change in thermal energy, in joules (J);
• $m$ is the mass, in kilograms (kg);
• $c$ is the specific heat capacity, in joules per kilogram degree Celsius (J/kg °C);
• is the , in degrees Celsius (°C).

Specific heat capacity tells you how much energy is needed to raise the temperature of 1 kg of a substance by 1 °C. A high value of $c$ means that a material can absorb a large amount of energy while undergoing only a small temperature rise. Water, for example, has a much higher specific heat capacity than most metals, so equal masses of water and metal do not warm by the same amount when given the same energy.

The units show why the equation produces energy:

$$\text{kg}\times\frac{\text{J}}{\text{kg}\,°\text{C}}\times °\text{C}=\text{J}$$

The kilograms and degrees Celsius cancel, leaving joules.

Finding the temperature change

The symbol $\Delta$ means “change in”, so calculate

$$\Delta\theta=\theta_{\text{final}}-\theta_{\text{initial}}.$$

For a rise from 18 °C to 43 °C, the temperature change is 25 °C, not 43 °C. A temperature interval has the same numerical size in degrees Celsius and kelvin: a rise of 25 °C is also a rise of 25 K.

When a substance cools, the calculation may give a negative $\Delta Q$ if changes are treated as signed quantities. This indicates that its thermal energy store has decreased and energy has been transferred away. If a question asks only for the amount of energy transferred, give the positive magnitude and state that the energy is transferred from the substance.

Rearranging the relationship

The same equation can be used to find any missing quantity:

$$c=\frac{\Delta Q}{m\Delta\theta}$$ $$m=\frac{\Delta Q}{c\Delta\theta}$$ $$\Delta\theta=\frac{\Delta Q}{mc}$$

These forms also reveal the proportional relationships. For a fixed material and temperature change, doubling the mass doubles the required energy. For a fixed mass and energy transfer, a larger specific heat capacity produces a smaller temperature change.

Before substituting, make the units consistent. In particular, convert grams to kilograms using

$$m\text{ in kg}=\frac{m\text{ in g}}{1000}.$$

When this equation applies

Use $\Delta Q=mc\Delta\theta$ when energy transfer produces a temperature change without a change of state. During melting or boiling, energy can be transferred while the temperature remains constant; that situation is described using specific latent heat instead. In a real heating experiment, some supplied energy may also transfer to the container and surroundings, so not all of it necessarily heats the material being studied.

Equation

$$\Delta Q=mc\Delta\theta$$

• $\Delta Q$: change in thermal energy (J)
• $m$: mass (kg)