Thermal expansion : linear and volumetric

THERMAL EXPANSION:

“The increase in the size of a substance on heating is called thermal expansion”.

The molecules of materials are always vibrating and moving. When materials are heated, the molecules vibrate with greater amplitudes due to increase in their kinetic energy. As a result the substance occupy more space. This is called expansion.

TYPES OF THERMAL EXPANSION:

There are two types of thermal expansions:

1. Linear Expansion

2. Volumetric Expansion

LINEAR EXPANSION:

“ The increase in length of a solid object upon heating is called linear expansion”.

It is one dimensional expansion as it occurs only along the length of the object

Mathematically it can be represented as:

$\boxed{\Delta L = \alpha L \Delta T}$

$L=$ Original Length of the object

$\Delta T=$ Change in Temperature

$\alpha =$ Coefficient of Linear expansion

VOLUMETRIC EXPANSION:

“The increase in volume of a solid object on heating is called volumetric expansion”.

It is three dimensional expansion as it occurs along the length, width and height of the object. Liquids and gases also undergo volumetric expansion.

Mathematically it can be represented as:

$\boxed{\Delta V = \beta V \Delta T}$

$V=$ Original Volume of the object

$\Delta T=$ Change in Temperature

$\beta =$ Coefficient of Volumetric expansion

RELATIONSHIP BETWEEN $\alpha$ AND $\beta$:

As linear expansion occurs in one dimension, where as volume expansion occurs in three dimensions. Hence, coefficient of volumetric expansion “$\beta$” is three times than

coefficient of linear expansion “$\alpha$”:

$\therefore \boxed{\beta= 3\alpha}$

DERIVATION FOR LINEAR EXPANSION:

Consider a thin uniform metallic rod of original length $L_o$. When it is heated through a small temperature $\Delta T$, its undergoes linear expansion and the final length of rod becomes $L$. The change in the length of rod $\Delta L$ is directly proportional to the initial (original) length of red and the rise in temperature. Hence,

$\Delta L \propto L_o$ ………… (i)

$\Delta L \propto \Delta T$ ………… (ii)

Combine equations (i) and (ii), We get:

$\Delta L \propto L_o\,\Delta T$

$\boxed{\Delta L =\alpha L_o\,\Delta T}$ …………(A)

Here, $\alpha$ is constant of proportionality and is called ‘Coefficient of linear expansion’.

Since,

$\Delta L = L -L_o$

Hence, equation (A) becomes:

$L -L_o =\alpha L_o\,\Delta T$

$L =\alpha L_o\,\Delta T +L_o$

$L =L_o(\alpha \,\Delta T + 1)$

OR

$\boxed{L =L_o(1 +\alpha \,\Delta T )}$