Question

What are linear coefficient of expansion and volume coefficient of expansion? Obtain derivations for them.

Answer 1

The tightly screwed lids or sealed bottles are first put in hot water so that the lid expands. ¡n case of liquids, when the thermometer is put in a slightly warm water, the mercury in it rises. When the thermometer is taken out, mercury falls again. In case of gases, a balloon partially inflated in a room (cool) expands when put in warm water. Also, a fully inflated balloon when placed in cold water starts shrinking.

Most of the substances expand on heating while contract on cooling. Change in the temperature of a body causes change in the dimensions of the body. The increase in dimensions of a body because of increase in its temperature is known as thermal expansion. The change in length is called linear expansion. The increase in area is called area expansion. The expansion in volume is called volume expansion. When the substance is in the form of a long rod, then for small change in temperature (∆T), the fractional change in length (∆l/l₀) is directly proportional to ∆T.

\(\frac{\Delta l}{lo}\propto\Delta T\)

∴ ∆T = α l₀ ∆T

\(\therefore\ \alpha = \frac{\Delta l}{lo}\times \frac{1}{\Delta T}\ ...(3)\)

Where α is called the coefficient of linear expansion and it is a characteristic of the material.

From equation (3),

l₀ ∆T α = ∆l = l - l₀

⇒ l = l₀ + l₀α ∆T

or  l = l₀(1 + α ∆T) ......(4)

Now considering the fractional change in area \(\frac{\Delta A}{A}\); the coefficient of areal expansion or superficial expansion (ß) is

Now, considering the fractional change in volume (∆V/V), the coefficient of volume expansion ß is

\(\gamma = \left(\frac{\Delta V}{V}\right)\times\frac{1}{\Delta T}\ ...(6)\)

where V is the initial volume of the substance. The unit of ß and γ is K^-1.

is also a characteristic of the substance. Generally, it depends on the temperature. It becomes constant only at a high temperature. The value of γ is more for those metals whose melting point is low. The value of γ is least for solid, more than solids in liquids, maximum in gases.

From ideal gas equation, we know that

PV = nRT ............(7)

At constant pressure,

P ∆V = nR ∆T

or, ∆V = \(\frac{nR}{P}\Delta T\)

Now, we will establish the relation between α and γ.
Consider a rectangular cuboid of length l₁, l₂ and l₃ and on increasing the temperature by ∆T, the length changes to l'₁, l'₂ and l'₃ respectively. Then, according to equation (4),

l'₁ = l₁(1 + α ∆T) ..........(9)

l'₂ = l₂(1 + α ∆T) ............(10)

and l'₃ = l₃(1 + α ∆T) ...........(11)

Then, with increase in temperature ∆T we represent the volume as:

V + ∆V = l'₁ l'₂ l'₃

or V + ∆V = l₁(1 + α ∆T) x l₂(1 + α ∆T) x l₃(1 + α ∆T)

or, V + ∆V = l₁ l₂ l₃ (1 + α ∆T) ........(11)

or, V + ∆V = V (1 + 3α ∆T + 3α² ∆T₂ + α³ ∆T₃)

[∵ V = l₁ l₂ l₃ and (a + b)³ = a³ + b³ + 3ab(a + b)]

Neglecting the higher powers of α, i.e., α² and α³, we have

V + ∆V ≈ V(1 + 3 α ∆T)

or, V + ∆V ≈ V + 3α V ∆T

or, \(\frac{\Delta V/V}{\Delta T} ≈ 3α\)

Using equation (8), we have

\(\gamma = 3\alpha\ ....(12)\)

This means that the coefficient of volume expansion is thrice the coefficient of linear expansion.

Similarly we can say ß = 2α

The anomalous behaviour of water:

Water shows unsual behaviour or anomalous behaviour when heated from 0°C to 4°C. Water does not expand when heated from 0°C to 4°C. If the temperature of water is increased beyond 4°C, then it expands. The variation of volume of water with increase in temperature is shown in figure.

So, water has maximum density at 4°C. This property of water has an important effect on the environment. Lakes and ponds freeze at the top first. A lake starts cooling around 4°C, water near to the surface losses energy and becomes denser and sinks. The warmer water near the bottom rises. When the colder water on the top reaches below 4°C, then it becomes less dense. This way, animal and plant life is saved in cold region.

Gases, at ordinary temperature, expand more than solids and liquids. For liquids, the coefficient of volume expansion is relatively independent of the temperature. However, for gases it is dependent as temperature. For an ideal gas, the coefficient of volume expansion at constant pressure can be found from the ideal gas equation.