Class 9 > Unit # 09:Thermal Properties of Matter > Linear & Volumetric Thermal Expansion

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Thermal Expansion: Linear and Volumetric Expansion

Q. What is Thermal Expansion? Describe linear expansion and volumetric expansion.

Thermal Expansion

"Whenever we supply some amount of heat to any solid substance, its dimensions increase. The increase in the size of an object due to heating is known as thermal expansion."

There are three main types of thermal expansion:

1. Linear Expansion: The expansion in length due to heating.
2. Superficial Expansion: The expansion in surface area due to heating.
3. Volumetric Expansion: The expansion in total volume due to heating.

1. Linear Expansion

"Whenever we supply heat to a rod, its length increases. This increment in length due to heating is known as linear expansion."

The linear expansion is represented by $\Delta L$ and is directly proportional to the original length ($L$) and the change in temperature ($\Delta T = T_2 - T_1$).

Mathematical Derivation

$\Delta L \propto L \quad \text{--- (i)}$
$\Delta L \propto (T_2 - T_1) \quad \text{--- (ii)}$

Combining equations (i) and (ii):

$\Delta L = \alpha L (T_2 - T_1)$

Where $\alpha$ is the constant of proportionality and is known as the coefficient of linear expansion.

If $L'$ represents the total length of the rod after heating:

$L' = L + \Delta L$
$L' = L + \alpha L (T_2 - T_1)$
$L' = L [1 + \alpha (T_2 - T_1)]$

Fig: Linear expansion of a solid rod upon heating from temperature $T_1$ to $T_2$.

2. Volumetric Expansion

"Whenever we supply heat to a solid cube, its volume increases. The increment in volume due to heating is known as volumetric expansion."

The volumetric expansion is represented by $\Delta V$ and is directly proportional to the original volume ($V$) and the difference in temperature ($(T_2 - T_1)$).

Mathematical Derivation

$\Delta V \propto V \quad \text{--- (i)}$
$\Delta V \propto (T_2 - T_1) \quad \text{--- (ii)}$

Combining equations (i) and (ii):

$\Delta V = \beta V (T_2 - T_1)$

Where $\beta$ is the constant of proportionality and is known as the coefficient of volumetric expansion.

If $V'$ represents the total volume after heating:

$V' = V + \Delta V$
$V' = V + \beta V (T_2 - T_1)$
$V' = V [1 + \beta (T_2 - T_1)]$

Fig: Volumetric expansion of a solid cube in three dimensions upon heating.

Relationship Between $\beta$ and $\alpha$

Since linear expansion occurs in one dimension, whereas volume expansion occurs in three dimensions (length, width, and height), the coefficient of volume expansion $\beta$ is three times the coefficient of linear expansion $\alpha$:

$\beta = 3\alpha$

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