Temperature and internal energy

The particles in an object move and interact with one another. Their motion gives them kinetic energy, while their positions and interactions give them potential energy. The total kinetic and potential energy of all the particles is the object’s internal energy.

Temperature is related to the average kinetic energy of the particles. A higher temperature means that the particles have greater average kinetic energy. When an object’s temperature rises, its particles gain kinetic energy, so its internal energy increases.

Temperature and internal energy are different

Temperature describes the average particle kinetic energy; internal energy is a total that also includes particle potential energy. Internal energy therefore depends on more than temperature:

Amount of material: At the same temperature, a larger amount of the same material in the same state has more particles and more internal energy.
State of the material: The arrangement and interactions of particles differ between states, affecting their potential energy. Internal energy can change during a change of state even when temperature remains constant.

For example, two samples of the same material can have the same temperature but different internal energies if they contain different amounts of material.

Specific heat capacity

The energy needed to raise an object’s temperature depends on its mass, the temperature increase and the substance it is made from. A larger mass needs more energy for the same temperature increase. Different substances also need different amounts of energy for the same mass and temperature increase.

The specific heat capacity of a substance is the energy required to raise the temperature of one kilogram of the substance by one degree Celsius. A substance with a higher specific heat capacity needs more energy per kilogram for the same temperature increase.

c = \frac{\Delta E}{m\Delta\theta}

In this equation:

$$$c$$$ is the specific heat capacity, in $\mathrm{J,kg^{-1},^{\circ}C^{-1}}$ .
$\Delta E$ is the energy transferred to the substance, in joules ( $\mathrm{J}$ ).
$$$m$$$ is the mass, in kilograms ( $\mathrm{kg}$ ).
$\Delta\theta$ is the temperature increase, in degrees Celsius ( $\mathrm{^{\circ}C}$ ).

The unit $\mathrm{J,kg^{-1},^{\circ}C^{-1}}$ means joules per kilogram per degree Celsius. Calculate the temperature increase by subtracting the initial temperature from the final temperature.

Using the specific heat capacity equation

The equation can be rearranged to find any one of the four quantities when the other three are known:

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

For example, suppose a $2,\mathrm{kg}$ object has a specific heat capacity of $500,\mathrm{J,kg^{-1},^{\circ}C^{-1}}$ . To raise its temperature by $10,\mathrm{^{\circ}C}$ , the energy transferred to it is:

\Delta E = 2 \times 500 \times 10 = 10{,}000\,\mathrm{J}

Measuring specific heat capacity

Specific heat capacity tells us how much energy is needed to raise the temperature of a unit mass of a material by one degree. To measure it, transfer a known amount of energy to a known mass of the material and measure the resulting temperature increase.

Calculate the specific heat capacity using:

c = \frac{\Delta E}{m\Delta\theta}

Here, $$$c$$$ is specific heat capacity in $\mathrm{J,kg^{-1},°C^{-1}}$ , $\Delta E$ is the energy transferred to the material in joules, $$$m$$$ is its mass in kilograms, and $\Delta\theta$ is its temperature increase in degrees Celsius.

Measuring a solid

Measure the mass of a metal block. Place an electrical heater and a thermometer in holes in the block, and record its initial temperature. Supply a measured amount of electrical energy through the heater, then record the final temperature. Subtract the initial temperature from the final temperature to find $\Delta\theta$ .

Measuring a liquid

Measure the mass of the liquid in an insulated container. Immerse an electrical heater and thermometer in the liquid, and record its initial temperature. Supply a measured amount of electrical energy, then record the final temperature and calculate the temperature increase. Stir the liquid so that its temperature remains as uniform as possible.

Measuring the electrical energy supplied

A joulemeter can measure the electrical energy supplied to the heater. Alternatively, if the heater’s power is known, calculate the energy supplied from the heating time:

E = Pt

Here, $$$E$$$ is energy in joules, $$$P$$$ is power in watts, and $$$t$$$ is time in seconds.

Effect of energy transfers

Insulation reduces energy transfer to the surroundings, but some supplied energy may still escape or heat the apparatus rather than the material. The electrical energy supplied is therefore not necessarily equal to the energy that raises the material’s temperature.

If all the supplied energy is used as $\Delta E$ in the calculation, even though some heats the apparatus or surroundings, the calculated specific heat capacity will be too high.