Thermal Physics (AQA A-Level Physics): Revision Notes

Specific heat capacity

Definition and concept

Specific heat capacity is the energy needed to raise the temperature of 1 kg of a material by 1°C without any change of state. This property is measured in joules per kilogram per degree Celsius (J kg⁻¹ °C⁻¹) or equivalently J kg⁻¹ K⁻¹.

Different materials require different amounts of energy to change their temperature. When energy is transferred to a substance through heating or work, and this results in a temperature rise, it indicates an increase in the kinetic energy of the particles. However, if expansion or a change of state (such as melting or vaporization) occurs, this indicates an increase in the potential energy component of the substance's internal energy instead.

Specific heat capacity values

The table below shows specific heat capacity values for common materials:

Material	Specific heat capacity (J kg⁻¹ °C⁻¹)
Air	993
Water	4190
Copper	385
Concrete	3350
Gold	135
Hydrogen	14300

Understanding the relationship

The greater the specific heat capacity of a material, the smaller the temperature change for a particular amount of thermal energy transferred. Consider what happens when energy $Q$ is transferred to a mass $m$ of a particular material, resulting in a temperature rise of $\Delta\theta$. The specific heat capacity $c$ can be determined using:

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

This equation can be rearranged to give the more commonly used form:

$Q = mc\Delta\theta$

Where:

• $Q$ = energy transferred (J)
• $m$ = mass of the substance (kg)
• $c$ = specific heat capacity (J kg⁻¹ °C⁻¹)
• $\Delta\theta$ = temperature change (°C or K)

Significance of water's high specific heat capacity

Water has a particularly large specific heat capacity of 4190 J kg⁻¹ °C⁻¹ compared with most other materials. This means that a relatively large amount of energy must be transferred to (or from) water to significantly raise (or lower) its temperature. When heated or cooled, the temperature of a large mass of water changes very slowly.

Determination of specific heat capacity by electrical method

The specific heat capacity of a solid can be found using an electrical heating method. In this approach, an immersion heater is placed in a cavity within a block of the material, and a thermometer is placed in another cavity. The block is heated for a measured time and the temperature rise is recorded.

The electrical power $P$ of the immersion heater is given by:

$P = VI$

Where $V$ represents the voltmeter reading and $I$ represents the current reading from the ammeter. The energy supplied to the block is $Q = VIt$, where $t$ is the time for which the power supply is switched on.

Assuming that all the energy supplied to the immersion heater is transferred to the material block, we can write:

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

Important considerations for accuracy

When measuring quantities in this experiment, various sources of uncertainty must be considered:

• The mass can be measured with a balance
• The temperature rise requires two temperature measurements which must be subtracted
• The voltage and current can be measured with a voltmeter and ammeter
• The heating time must be measured, with uncertainty depending on the experimenter's reaction time

Determination of specific heat capacity by method of mixtures

The specific heat capacity of a solid object can be determined by an alternative method called the method of mixtures. This involves heating the solid object in boiling water to bring its temperature to 100°C. The object is then quickly transferred to an insulated beaker containing water at room temperature, and the temperature of the mixture is monitored with a thermometer.

The data required for this method are:

• The mass of the object ($m_{\text{object}}$)
• The mass of water in the beaker ($m_{\text{water}}$)
• The volume of water ($V$)
• The initial temperature of the water ($\theta_1$)
• The final maximum temperature of the mixture ($\theta_2$)

The energy transferred from the heated object to the cool water is:

$Q = m_{\text{object}}c_{\text{object}} \times (100 - \theta_2)$

Assuming that all the internal energy lost by the object is transferred to the water and all the energy gained by the water is used to raise its temperature, the energy gained by the water is:

$Q = m_{\text{water}}c_{\text{water}} \times (\theta_2 - \theta_1)$

Equating these two expressions for $Q$ gives:

$m_{\text{object}}c_{\text{object}} \times (100 - \theta_2) = m_{\text{water}}c_{\text{water}} \times (\theta_2 - \theta_1)$

The mass of water is related to its volume by $m_{\text{water}} = \rho V$, where $\rho$ is the density of water (1000 kg m⁻³). The specific heat capacity of water is $c_{\text{water}}$ = 4190 J kg⁻¹ °C⁻¹. This equation can be rearranged to determine the specific heat capacity of the object:

$c_{\text{object}} = \frac{1000 \times V \times 4190 \times (\theta_2 - \theta_1)}{m_{\text{object}} \times (100 - \theta_2)}$

Increasing internal energy by doing work

When an object of mass $m$ is dropped from a height $h$ and hits the ground without bouncing, all of its kinetic energy on reaching the ground is transferred as internal energy. The temperature of the object rises by an amount dependent on its mass and its specific heat capacity.

The energy transferred is equal to the final kinetic energy of the object, which equals the gravitational potential energy gained when raising it to height $h$:

$\text{work done} = mgh$

The experiments conducted by James Joule on the equivalence of heating and working showed that the energy transferred to a substance by doing work, resulting in a temperature rise $\Delta\theta$, is equal to $mc\Delta\theta$. Therefore:

$mgh = mc\Delta\theta$

This simplifies to give the temperature rise:

$\Delta\theta = \frac{gh}{c}$

Determination of specific heat capacity by calculating work done

An estimate for the specific heat capacity of lead can be determined using an inversion tube method. Lead shot of total mass $m$ is contained in a cardboard tube. The tube, of length $h$, has a rubber bung at one end and a second rubber bung with a hole through its centre at the other end.

The initial temperature of the lead is measured by inserting a thermometer through the hole in the bung. The tube is held vertically and then repeatedly turned end to end, so that the lead is continually being lifted and allowed to fall to the other end of the tube. After $N$ inversions, the internal energy of the lead shot increases because work is done on the falling lead shot to bring it to rest as it hits the bung at the bottom of the tube.

On inverting the tube, the lead shot gains an additional amount of gravitational potential energy equal to $mgh$. As the lead shot falls, its gravitational potential energy is converted to kinetic energy, and the work done by the bung to bring the lead shot to rest is equal to that initial gravitational potential energy, $mgh$. If the total number of inversions is $N$, the total work done on the lead shot is $Nmgh$.

The final temperature of the lead is measured by replacing the glass rod with a thermometer, enabling the temperature rise $\Delta\theta$ to be determined. Assuming that all the work done, $Nmgh$, is converted to the internal energy of the lead shot:

$c = \frac{\text{energy transferred}}{m\Delta\theta} = \frac{Nmgh}{m\Delta\theta} = \frac{Ngh}{\Delta\theta}$

Flow calculations

Because of its high specific heat capacity, water is effective for storing internal energy and also for transporting it. Water is used as the fluid in the cooling systems of many engines and other machinery. Its high specific heat capacity means that a lot of excess thermal energy can be removed from the engine without the water temperature becoming too high. Energy transfer calculations often need to be performed involving a flow of water.