# Expansion of Liquids

**Liquids just like solids**expand on heating and contract on cooling. The expansion and contraction in liquids is**more than**those in solids.- The reason for this is that the
**intermolecular force in**liquids is less than that in solids. - Hence, the molecules in a liquid
**vibrate more than**the molecules in a solid when they**get thermal energy**. **Since the liquids do not have a shape**of their own but possess a fixed volume, we have to consider their**co-efficient of cubical expansion**while studying their expansion.**When a liquid is heated**in a vessel, the vessel also expands on heating. Therefore,**while studying**the cubical expansion of liquids,**the expansion of container**should also be taken into account.- The reading of expansions of liquids would have been a little more, had there been no expansion of the vessel.
**So in case of liquids**, we define**two coefficients of cubical**expansion.

**Real and apparent expansion of a liquid**

- The expansion of the liquid due
**to increase in temperature**is always associated by**expansion of containing vessel.** - The expansion which we observe is called
**apparent expansion**which is always less than**real expansion of the liquid**. - Let us take a test tube filled with coloured water. Its mouth is closed with a rubber cork having a hole.
**A capillary tube**is inserted into the test tube through the hole in the cork as in figure.

**Initially, the water rises** in the capillary tube. Let the water rises up-to **point A**. The test tube is placed in a beaker in which water is boiled. Initially, the water level in the capillary falls. Let it falls to **point B**. After that, water level begins rising and it rises above the initial level. Let it rises up-to **point C**.

**The fall in water** level in the capillary **in the beginning of the heating** is due to the **expansion of the capillary**. It is because **the capillary tube which** is made up of glass expands more rapidly than water. So in the beginning, the capillary expands but the water does not. Therefore, **we observe fall in water level**.

**After sometime**, the water in the capillary **also gets enough heat** to expand. When the water starts expanding, **the water level rises** in the capillary. Since the expansion of the water in the capillary **is more** than the expansion of the capillary, the water level **rises above the initial level** in the capillary.

**From this experiment**, it is clear that **a liquid like water** expands on heating. This experiment shows that

- Volume of water from
**A to B in**the capillary represents**expansion of the capillary.** - Volume of water from
**A to C**in the capillary represents**apparent (or observed) expansion of water.**

Hence, the real expansion of water is given by

**Real expansion** = expansion of the capillary + apparent expansion of water

**For any liquid,**

Real expansion of liquid = expansion of vessel + apparent expansion of liquid

**Coefficient of real expansion (γ _{r})**

- It is defined as
**the real increase in volume**of the liquid per unit volume**per unit rise**in temperature. - That is,
**γ**= Real increase in volume/( Original volume × rise in temperature)_{r}

Let **V _{0}** and

**V**be the volumes of a liquid

_{θ}**at 0**and at

^{o}C**θ**respectively, then

^{o}C**γ _{r }= (V_{θ }– V_{0})/( V_{0 }×θ)**…………………………… (i)

**V _{θ }= V_{0 }(1 + γ_{r} θ)**…………………………………… (ii)

**Coefficient of apparent expansion (γ _{a})**

- It is defined as
**the apparent (or observed) increase in volume**per unit volume of the liquid per unit rise of temperature. - That is,
**γ**= Apparent increase in volume/ (Original volume × rise in temperature)_{a}

**Relation between Coefficient of Real and Apparent Expansion**

- Let us consider some liquid contained in a capillary tube at room temperature
**θ**with level of liquid_{1}^{o}C**at A**as shown in figure. Let**V**be the original volume of the liquid. Suppose the system is heated to temperature**θ**. As heat is supplied to the system,_{2}^{o}C**the capillary expands first**and the level of liquid falls to the**level B**. Thus, volume**ΔV**in between levels_{g}**A**and**B**gives the expansion of the capillary tube. Therefore,

**ΔV _{g }= V γ_{g}(θ_{2 }– θ_{1})**………………………………… (iii)

where **‘γ _{g}’** is coefficient of cubical expansion of the glass.

**On heating**, the liquid expands and its level moved up from **B to C**.

Thus, the real increase in volume **ΔV _{r} **is equal to the level between

**B and C**.

Therefore,

** ΔV _{r} = V γ_{r}(θ_{2 }– θ_{1})**………………………………… (iv)

where** ‘γ _{r}’** is the coefficient of real expansion of liquid.

Apparently, the liquid in the capillary tube rises from **A to C**.

Thus, the apparent increase in volume is given by,

**ΔV _{a} = V γ_{a}(θ_{2 }– θ_{1})**………………………………….. (v)

where **‘γ _{a}’** is the coefficient of apparent expansion of liquid.

From the figure it is clear that,

**BC = AB + AC**

**ΔV _{r} = ΔV_{g} + ΔV_{a}**

**V γ _{r}(θ_{2 }– θ_{1}) = V γ_{g}(θ_{2 }– θ_{1}) + V γ_{a}(θ_{2 }– θ_{1})**

**γ _{r} = γ_{g }+ γ_{a} **…………………………………………….. (vi)

Thus, the **coefficient of real expansion** of the liquid is equal to **the sum of the coefficient** of cubical expansion of the container and the coefficient of apparent expansion of the liquid.

If **‘α’** is the coefficient of **linear expansion** of the material of the containing vessel,

Then, **γ _{g} = 3α**

The equation (vi) becomes

**γ _{r} = 3α + γ_{a}**……………………………………………… (vii)

**References: **

i) https://en.wikipedia.org/wiki/Thermal_expansion