Specific Heat Capacity Calculations

Specific heat capacity is the energy required to raise the temperature of 1 kg of a substance by 1 $^\circ$C (or 1 K).

The equation

$\Delta E = mc\Delta\theta$

Symbol	Quantity	Unit
$\Delta E$	Energy transferred	J
$m$	Mass	kg
$c$	Specific heat capacity	J/(kg $^\circ$C)
$\Delta\theta$	Temperature change	$^\circ$C or K

Key facts

• Different materials have different specific heat capacities. Water has a high specific heat capacity (4200 J/(kg $^\circ$C)), meaning it takes a lot of energy to heat.
• The equation works for both heating and cooling. When cooling, $\Delta\theta$ is the temperature drop and $\Delta E$ is the energy released.

Worked example

A 0.40 kg metal block receives 9600 J and its temperature rises by 30 $^\circ$C. Calculate its specific heat capacity.

$c = \frac{\Delta E}{m\Delta\theta} = \frac{9600}{0.40 \times 30} = 800\text{ J/(kg }^\circ\text{C)}$

Worked example: finding energy

How much energy is needed to heat 2.0 kg of water from 20 $^\circ$C to 100 $^\circ$C? ($c_{\text{water}} = 4200$ J/(kg $^\circ$C))

$\Delta\theta = 100 - 20 = 80\text{ }^\circ\text{C}$

$\Delta E = 2.0 \times 4200 \times 80 = 672\,000\text{ J} = 672\text{ kJ}$

Experimental method

1. Measure the mass of the substance.
2. Record the initial temperature.
3. Heat using an electrical heater of known power for a measured time.
4. Record the final temperature.
5. Calculate energy supplied: $E = Pt$.
6. Calculate $c$ from the equation.

Sources of error: energy lost to surroundings (makes the calculated $c$ too high because not all energy heats the block), and using insufficient insulation.

Common errors and how to correct them

Using the final temperature instead of the temperature change. $\Delta\theta$ is the difference between the final and initial temperatures, not the final temperature alone.

Forgetting to convert grams to kilograms. The unit of $c$ uses kg, so mass must be in kg.

How to apply this in an exam

State the equation, convert units (kJ to J, g to kg), substitute, and calculate. If asked about the experiment, describe how to minimise heat loss (insulation, lid) and state the assumption that all electrical energy heats the block.