Explain the volumetric thermal expansion.
OR
Derive the relation for volume thermal expansion in solids.
OR
Show that V =$ V_{0} (1+ \beta \triangle T)$?
Difficulty: Medium
Volumetric thermal expansion:
The volume of a solid also changes with the temperature change and is called volume thermal expansion or cubical thermal expansion.
Consider a solid of the initial volume$ V_{0}$ at a certain temperature $ T_{0}$. On heating, the solid to a temperature T, let its volume becomes V, then
Change in the volume of a solid $ \triangle V = V - V_{0}$
And change in temperature $ \triangle T$ =$ T - T_{0}$
Like linear expansion, the change in $ \triangle V$ is found to be proportional to its original volume $ V_{0}$ and change in temperature $ \triangle T $. Thus
$ \triangle V \alpha V_0 \triangle T $
or $ \triangle V $ = $ \beta V_0 \triangle T $ ………. (i)
or $ V - V_{0} = 〖\beta V〗_{0} \triangle T$
or $V = V_{0}+\beta V_{0} \triangle T $
or $ V = V_{0} (1+ \beta \triangle T)$ ………. (ii)
where \beta is the temperature coefficient of volume thermal expansion, using equation (i), we get
$ \beta $ = $ (\triangle V )/(V_{0} \triangle T )$ (iii)
Coefficient of volume expansion $ \beta$:
Thus, we can define the temperature coefficient of volume expansion $ \beta $ as the fractional change in its volume per kelvin change in temperature. The coefficient of linear expansion and volume expansion are related by the equation:
$\beta = $3 \alpha$ …………….. (iv)
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