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# Heat capacity and latent heat

## Heat capacity

The heat capacity ($C$) of an object is the amount of thermal energy required to raise its temperature by one degree Celsius (or equivalently by one kelvin).

The heat capacity is given by:$$\Torange{\text{heat capacity}}=\frac{\Tred{\text{thermal energy}}}{\Tblue{\text{change in temperature}} } \quad \Torange{C} = \frac{\Tred{Q}}{\Tblue{\Delta T}}$$

Heat capacity is measured in joules per degree Celsius (or kelvin): $\text{J} / ^{\circ} \text{C}$ (or $\text{J K}^{-1}$)

A container of water absorbs $\Tred{2000 \text{ J}}$ of thermal energy and heats up by $\Tblue{2^{\circ}\text{C}}.$

The heat capacity of the water is $\Torange{1000 \text{ J}/^{\circ}\text{C}}.$

Objects with higher heat capacities cool and warm more slowly than objects with lower heat capacities.

The heat capacity of one litre of air is about $1.3 \text{ J}/^{\circ}\text{C}.$

## Specific heat capacity

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

\begin{align*}\Torange{\text{specific heat capacity}}=&\frac{\Tred{\text{thermal energy}}}{\Tblue{\text{temperature change}} \times \Tviolet{\text{mass}} } \\ \Torange{c} =& \frac{\Tred{Q}}{\Tblue{\Delta T}\Tviolet{m}} \end{align*}

Heat capacity is measured in $\text{J}/ ^{\circ}\text{C} \text{ kg}$ (or $\text{J kg}^{-1}\text{K}^{-1}$)

$\Tviolet{2 \text{ kg}}$ of water absorbs $\Tred{8400 \text{ J}}$ of thermal energy and its temperature rises by $\Tblue{1^{\circ}\text{C}}.$

The specific heat capacity of water is: $$\Torange{c} = \frac{\Tred{8400 \text{ J}}}{\Tblue{1 ^{\circ}\text{C}} \times \Tviolet{2 \text{ kg}}} = \Torange{4200 \text{ J}/^{\circ}\text{C}\text{ kg}}$$

Specific heat capacity is a property of a material and does not depend on mass. Heat capacity is a property of an object and depends on the mass of the object.

The specific heat capacity of iron is $450 \text{ J}/^{\circ}\text{C}\text{ kg}$

## Latent heat

Latent heat ($L$) is the energy absorbed or released by a substance during a change of state at constant temperature.

When water is heated, its temperature increases up to the boiling point ($100^{\circ}\text{C}$).

Once the boiling point has been reached, the temperature remains constant even though additional energy is being supplied to the water.

This additional energy is used to convert the water into a vapour at the boiling point temperature.

The energy absorbed by the water during its conversion to vapour is called latent heat.

The SI unit of latent heat is the same as that of energy ($J$).

Latent heat is absorbed by ice when it melts into water

## Specific latent heat

The specific latent heat ($\ell$) of a substance is the thermal energy absorbed or released when $1 \text{ kg}$ of a substance changes state at constant temperature.

The specific latent heat is given by:$$\Torange{\text{specific latent heat}}=\frac{\Tred{\text{thermal energy}}}{\Tblue{\text{mass}}} = \frac{\Tred{Q}}{\Tblue{m}}$$

The SI unit of specific latent heat is joules per kilogram ($\text{J} / \text{kg}$).

$\Tblue{0.2 \text{kg}}$ of ice absorbs $\Tred{60\,000 \text{J}}$ as it melts into water.

The specific latent heat is $\dfrac{\Tred{60\,000 \text{J}}}{\Tblue{0.2 \text{ kg}}} = \Torange{300\, 000} \text{ J}/\text{kg}.$

Latent heat is absorbed by ice when it melts into water

## Types of latent heat

The latent heat of a substance depends on the change of state in question (e.g. solid to liquid).

• The latent heat of fusion ($L_{\text{f}}$) is the heat required to change the state of a substance from solid to liquid at constant temperature.

This is exactly equal to the heat released when the substance changes from a liquid to a solid.

• The latent heat of vaporisation ($L_{\text{v}}$) is the heat required to change the state of a substance from liquid to gas at constant temperature.

This is exactly equal to the heat released when the substance changes from a gas to a liquid.

These latent heat values have corresponding specific latent heat values ($\ell_{\text{f}}$ and $\ell_{\text{v}}$).

The specific latent heat of fusion of water is $\ell_{\text{f}}=334 \text{ kJ}/\text{kg}$ and the specific latent heat of vaporisation of water is $\ell_{\text{v}}=2260 \text{ kJ}/\text{kg}$.

The latent heat of fusion is the energy absorbed by ice when it melts into water

## Summary of heat capacity and latent heat

A summary of the properties of the four different measures of thermal energy is given below.

Heat capacity

• Thermal energy required to raise the temperature of an object by $1 ^{\circ}\text{C}$
• Depends on material and on mass
• $\Torange{C} = \dfrac{\Tred{\text{thermal energy}}}{\Tblue{\text{change in temperature}} } = \dfrac{\Tred{Q}}{\Tblue{\Delta T}}$
• Measured in $\text{J} / ^{\circ} \text{C}$ or $\text{J K}^{-1}$

Specific heat capacity

• Thermal energy required to raise the temperature of $1\text{ kg}$ of a material by $1 ^{\circ}\text{C}$
• Refers to a material and does not depend on mass
• $\Torange{c}=\dfrac{\Tred{\text{thermal energy}}}{\Tblue{\text{change in temperature}} \times \Tviolet{\text{mass}}} = \dfrac{\Tred{Q}}{\Tblue{\Delta T}\Tviolet{m}}$
• Measured in $\text{J} / ^{\circ} \text{C}\text{kg}$ or $\text{J kg}^{-1}\text{K}^{-1}$

Latent heat

• Energy released or absorbed when a substance changes state at constant temperature
• Depends on material and on mass
• Different latent heats for different phase changes
• Symbol is $\Torange{L}$, and is measured in $\text{J}$

Specific latent heat

• Energy released or absorbed when $1 \text{ kg}$ of a substance changes state at constant temperature
• Refers to a material and does not depend on mass
• $\Torange{\ell}=\dfrac{\Tred{\text{thermal energy}}}{\Tviolet{\text{mass}}} = \dfrac{\Tred{Q}}{\Tviolet{m}}$
• Measured in $\text{J} /\text{kg}$