Internal energy and specific heat capacity

Energy stored by particles

A system is the object or collection of objects being studied, such as a metal block or water in a beaker. Its atoms and molecules store energy. The total energy associated with their motion and their positions relative to one another is called internal energy.

Particles have kinetic energy because they move: particles in a solid vibrate, while particles in liquids and gases also move from place to place. They have potential energy because of their positions relative to other particles and the forces between them.

Heating transfers energy into a system and increases its internal energy. This can raise its temperature or cause a change of state. These are different effects of the same energy transfer.

Temperature rise or change of state?

When heating raises the temperature, the particles gain kinetic energy. They move or vibrate faster on average. Temperature is related to their average kinetic energy; it is not a measure of the total energy stored by all the particles.

During melting or boiling, energy can continue to enter the substance without raising its temperature. Instead, energy increases the particles’ potential energy as their arrangement changes and forces of attraction are overcome. Internal energy therefore increases even though the temperature remains constant during the change of state.

For example, warming solid ice towards its melting point raises its temperature. Melting the ice at its melting point requires further energy but does not raise its temperature. Specific heat capacity describes energy needed for a temperature rise; specific latent heat describes energy needed for a change of state without a temperature change.

Why materials warm by different amounts

The temperature rise depends on the energy supplied, the mass heated and the material.

• Supplying more energy to the same mass of the same material produces a larger temperature rise.
• Heating a larger mass by the same temperature change requires more energy.

The specific heat capacity of a substance is the amount of energy required to raise the temperature of one kilogram of the substance by one degree Celsius.

A material with a higher specific heat capacity needs more energy for the same temperature rise. If equal masses receive energy at the same rate, the material with the higher specific heat capacity warms more slowly.

For example, aluminium has a specific heat capacity of about 913 J/kg °C, while copper has a value of about 385 J/kg °C. Raising 1 kg of aluminium by 1 °C therefore requires more than twice as much energy as raising 1 kg of copper by 1 °C.

Calculating changes in thermal energy

For a temperature change without a change of state:

$$\Delta E = mc\Delta\theta$$

Here, $\Delta E$ is the change in thermal energy in joules (J), $m$ is mass in kilograms (kg), $c$ is specific heat capacity in J/kg °C, and $\Delta\theta$ is temperature change in degrees Celsius (°C).

Calculate the temperature change first:

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

Then multiply the mass, specific heat capacity and temperature change. The equation can also be rearranged to find an unknown quantity:

$$c = \frac{\Delta E}{m\Delta\theta} \qquad \Delta\theta = \frac{\Delta E}{mc}$$

The equation applies to cooling too. Cooling decreases the system’s thermal energy. If asked for the amount of energy transferred out, use the size of the temperature fall to give a positive amount; a signed change in thermal energy is negative.

Required practical: measuring specific heat capacity

An electrical heater can transfer a known amount of energy to a metal block. Measuring the block’s mass and temperature rise allows its specific heat capacity to be determined.

Use a metal block with holes for an immersion heater and thermometer, a balance, insulation, a stopwatch and a suitable low-voltage power supply. The heater’s power can be provided or determined using an ammeter and voltmeter.

1. Measure the block’s mass in kilograms using the balance.
2. Insert the heater and thermometer into their holes. A drop of water in the thermometer hole improves thermal contact by reducing the insulating air gap.
3. Wrap the block in insulation to reduce energy transfer to the surroundings. Record its initial temperature.
4. Switch on the heater and start the stopwatch. Keep the heater’s power constant. If measuring it, connect the ammeter in series with the heater and the voltmeter across it, then use $P=VI$.
5. Record the temperature at regular intervals, such as every minute for about ten minutes. Record the elapsed time for each reading.
6. Once a clear temperature rise and a useful set of readings have been obtained, switch off the heater. Calculate the energy supplied and the temperature rise, then determine $c$.

The electrical work done by a constant-power heater is:

$$E_{\text{supplied}} = Pt$$

Power is measured in watts and time in seconds, giving energy in joules. If losses are small, the energy supplied is approximately equal to the increase in the block’s thermal energy:

$$c \approx \frac{Pt}{m\Delta\theta}$$

With a battery supply, the battery’s chemical energy store decreases as electrical work is done. The block’s internal energy increases, producing the measured temperature rise. Some energy also warms the heater and the surroundings, so not all the energy supplied reaches the block.

Use a suitable low-voltage supply and keep water away from electrical connections. The block and heater can become hot enough to burn: do not touch them during heating, switch off before adjusting the apparatus, and allow them to cool before handling.

Making sense of the results

Insulation reduces energy losses but cannot eliminate them. Better thermal contact helps the thermometer follow the block’s temperature more closely. Repeating the investigation and comparing calculated values helps assess how consistent the results are.

A graph can use energy supplied on the horizontal axis and temperature on the vertical axis. Calculate the energy supplied at each recorded time using $Pt$. The graph may curve initially because the heater and block take time to warm and the thermometer reading can lag behind.

For the approximately straight section, the gradient is temperature rise divided by energy supplied. Since $\Delta\theta=\Delta E/(mc)$:

$$\text{gradient} = \frac{1}{mc}$$

A shallower gradient means a smaller temperature rise per joule. For blocks of equal mass, this indicates a higher specific heat capacity. Using a straight section of the graph also draws on several readings rather than relying on just one temperature measurement.

Internal energy

• Internal energy is stored by particles through their motion and relative positions.
• Heating increases internal energy: temperature may rise, or a change of state may occur.
• Temperature rise: average particle kinetic energy increases.
• Melting or boiling: particle potential energy increases while temperature remains constant.

Specific heat capacity

Energy needed to raise 1 kg of a substance by 1 °C.

$$\Delta E = mc\Delta\theta$$

Units: $\Delta E$ in J; in kg; in J/kg °C; in °C.