# Thermal Expansion, Heat and Phase Change

Despite the difference between heat and temperature, adding heat in a body increases its temperature and removing heat decreases its temperature. You may have noticed that every material expand when heated (added heat to it) other than the anomalous behaviour of some materials such as water discussed below. Neglecting other materials which contract on heating, as the temperature of a material increases, the dimensions of the material increase. You may notice that a tight lid can be loosen by pouring hot water on it.

### Linear Expansion

Linear expansion of a material is its expansion along one dimension. We consider an example of a rod and find the change in length of the rod due to expansion. The initial length of the rod is $L_i$ and the initial temperature is $T_i$. As the temperature of the rod increases, its final temperature is $T_f$ and final length is $L_f$ ($T_f>T_i$ and $L_f>L_i$). The rod undergoes linear expansion along its length. It has been experimentally found that the change in length $ΔL=L_f−L_i$ is directly proportional to the initial length $L_i$ and the change in temperature $ΔT=T_f−T_i$ that is, $\Delta L \propto {L_i}{\kern 1pt} \Delta T$ and

\[\Delta L = \alpha {L_i}{\kern 1pt} \Delta T \tag{1} \label{1}\]

where $\alpha$ is the proportionality constant called coefficient of linear expansion. The above equation is valid only when the temperature change is not too high otherwise $\alpha$ is not constant.

### Volume Expansion

Volume expansion is similar to linear expansion but in this case whole volume of the material expands not just in one dimension. The material we have undergoes a change in temperate $\Delta T$ and its volume increases. So, we consider initial volume $V_i$, initial temperature $T_i$, final volume $V_f$ and final temperate $T_f$. Similar to the linear expansion above, the change in volume $\Delta V = V_f - V_i$ due to the change in temperature $\Delta T = T_f - T_i$ is directly proportional to the initial temperature $V_i$ and the change in temperature $\Delta T$ that is, $\Delta V \propto {V_i}{\kern 1pt} \Delta T$ and

\[\Delta V = \beta {V_i}{\kern 1pt} \Delta T \tag{2} \label{2}\]

where $\beta $ is the proportionality constant called coefficient of volume expansion and it is constant when the temperature change is not too high.

For solids the coefficient of volume expansion $\beta$ is related to the coefficient of linear expansion by $\beta = 3\alpha $. Can you prove this? Consider a cube of initial length $l$ whose volume $V$ is $l^3$. Then, $dV = 3{l^2}dl$. You know from Eq. \eqref{1} that $dl = \alpha l{\kern 1pt} dT$, so you can write $dV = 3{l^2}\alpha l{\kern 1pt} dT = 3{l^3}\alpha {\kern 1pt} dT = 3V\alpha {\kern 1pt} dT$:

\[\frac{dV}{{VdT}} = 3\alpha \]

From Eq. \eqref{2}, $\beta = \frac{{dV}}{{VdT}}$ and hence

\[\beta = 3\alpha \tag{3} \label{3}\]

### Anomalous Behaviour of Water

You know how much important matter the water is for life. But you might not know its special property. Most materials expand when heated or when their temperature is increased but water contracts when heated between $0{{\kern 1pt} ^ \circ }{\rm{C}}$ and $4{{\kern 1pt} ^ \circ }{\rm{C}}$. And above $4{{\kern 1pt} ^ \circ }{\rm{C}}$ it again expands. Water shows its anomalous behaviour only between the temperatures $0{{\kern 1pt} ^ \circ }{\rm{C}}$ and $4{{\kern 1pt} ^ \circ }{\rm{C}}$. Since the volume of water decreases continuously from $0{{\kern 1pt} ^ \circ }{\rm{C}}$ to $4{{\kern 1pt} ^ \circ }{\rm{C}}$, it has the maximum density at $4{{\kern 1pt} ^ \circ }{\rm{C}}$. You can see the ice floating on the surface of ponds or lakes. Because water at $4{{\kern 1pt} ^ \circ }{\rm{C}}$ has the greatest density, it continuously flows towards the bottom of a lake. This has a very useful effect on animals who still can survive at the bottom of the lake even if the surface of the lake is frozen. This is the reason why ponds and lakes start to freeze from the surface not from the bottom.

## Heat and Phase Change

The change in temperature is accompanied by the heat transfer and we have already noted in this article that the heat is the form of energy transferred from one body to another. Heat is the energy in *transit* but a body has its own *heat capacity*.

Let $Q$ denotes the quantity of heat. The change in temperature in a body of mass $m$ is $\Delta T$. It has been experimentally found that the quantity of heat is directly proportional to the mass $m$ and the change in temperature $\Delta T$, that is, $Q \propto m\Delta T$. The quantity of heat if $c$ is the proportionality constant is given by

\[Q = mc\Delta T \tag{4} \label{4}\]

where $c$ is called *specific heat* which is

\[c = \frac{Q}{{m\Delta T}}\]

and can be defined as the amount of heat required to raise the temperature of $1\text{Kg}$ material by $1\text{K}$ (or $1{{\kern 1pt} ^ \circ }{\rm{C}}$). Its unit is ${\rm{J/(Kg}} \cdot {\rm{K)}}$. Note that the SI unit of the quantity of heat is $J$ same as the unit of mechanical energy. Mechanical energy and heat energy both are the forms of energy which can be converted to each other; mechanical energy can be converted to heat energy and vice versa.

You'll often encounter another unit of the quantity of heat called Calorie abbreviated as $\text{Cal}$. $1$ Calorie is the amount of heat required to raise the temperature of $1\text{g}$ of water from $14.5{{\kern 1pt} ^ \circ }{\rm{C}}$ to $15.5{{\kern 1pt} ^ \circ }{\rm{C}}$. Experiments have shown that

\[{\rm{1Cal}} = 4.186{\kern 1pt} {\rm{J}} \]

The Eq. \eqref{4} can also be expressed in terms of *molar specific heat* denoted by $C$. If $M$ is the molar mass of a material and $n$ is the total number of moles, the total mass of the material is $m = nM$ and

\[Q = nMc\Delta T = nC\Delta T \tag{5} \label{5}\]

where the product $Mc$ is constant called molar specific heat $C$ which is the amount of heat required to raise the temperature of one mole of the material by $1\text{K}$ (or $1{{\kern 1pt} ^ \circ }{\rm{C}}$).

### Phase Change

In the universe you can see different states of the same substance. A particular state of a substance is called phase. For example, water can exist as solid (ice), liquid and gas (vapour) states. Therefore, solid, liquid and gaseous forms of water are different phases of water such as solid phase, liquid phase and gaseous phase. The phase change takes place when one phase of a substance changes into other phase (for example the state of water can change from liquid phase to solid phase etc.). The change in phase is called *phase transition*.

In phase transition, one phase of a material can change into another phase by absorbing or releasing a certain quantity of heat without any change in temperature. For example, ice at $0{{\kern 1pt} ^ \circ }{\rm{C}}$ is converted into liquid at $0{{\kern 1pt} ^ \circ }{\rm{C}}$ by absorbing a certain quantity of heat $Q$. We introduce a new heat called *latent heat* which is the amount of heat per unit mass required to turn one phase of a substance into another phase without changing its temperature. The quantity of heat per unit mass required to change the phase of a substance from solid to liquid is called *latent heat of fusion* and the transition is solid-liquid phase transition. If $m$ is the mass of a substance in its solid form and $L_f$ is the latent heat of fusion, the quantity of heat which must be absorbed by the substance to turn into liquid at the same temperature as in solid form is

\[Q = {L_f}m \]

In the reverse process that is when the liquid turns into solid,` the same quantity of heat given by the above expression is released by the substance and the quantity of heat in this case is negative:

\[Q = -{L_f}m\]

Note that in phase transition a certain amount of heat is used by the substance to change the phase of the substance not to increase or decrease the temperature of the substance. That means the ice at $0{}^ \circ {\rm{C}}$ changes into liquid at $0{}^ \circ {\rm{C}}$ by absorbing a certain quantity of heat. As the solid form of the substance is melting into liquid form, both forms are at the same temperature; the melting temperature is the same as the freezing temperature and the melting and freezing processes are accompanied by absorbing or releasing a certain amount of heat without increasing or decreasing the temperature of the substance respectively. But yes if you supply heat greater than a substance needs to turn completely into another phase, you increase the temperature of the substance after the substance turns completely into another phase. The two phases of a substance at the same temperature is called *phase equilibrium*. Combining both above expressions for heat required to change the phase gives:

\[Q = \pm {L_f}m \tag{6} \label{6}\]

Similar to the latent heat of fusion there is *latent heat of vaporization* which is the quantity of heat per unit mass required to change liquid phase to vapour (gas) phase without changing temperature. The transition is liquid-gas phase transition. In the reverse process of liquid-gas phase transition the gaseous phase turns into liquid phase releasing the same quantity of heat absorbed by the liquid to turn into gas. For example, water boils at $100{}^ \circ {\rm{C}}$ and turns into vapour. Liquid water at $100{}^ \circ {\rm{C}}$ absorbs a certain quantity of heat depending on its mass to turn into steam at $100{}^ \circ {\rm{C}}$ and in the reverse process the steam condenses or turns into liquid water at $100{}^ \circ {\rm{C}}$ releasing the same amount of heat absorbed before. You saw in phase transition that a certain quantity of heat is absorbed or released only to change the phase not the temperature. If $L_v$ is the latent heat of vaporization and $m$ is the mass of a substance, the quantity of heat in both processes (as in Eq. \eqref{6}) required for the phase change is:

\[Q = \pm {L_v}m \tag{7} \label{7}\]

In some cases, the solid form of a substance turns directly into gaseous form and this process is called sublimation. The latent heat of sublimation is denoted similarly by $L_s$.