Question

The ratio $\alpha :\beta :\gamma $ (where $\alpha ,\beta $ and $\gamma $ are coefficient of linear thermal expansion, coefficient of areal thermal expansion and coefficient of volumetric thermal expansion respectively) is

Answer 1

- Hint:You can start by defining what thermal expansion is, follow it up explaining linear expansion, area expansion and volume expansion with their respective equations, i.e. $\dfrac{{\Delta L}}{L} = \alpha \Delta T$ , $\dfrac{{\Delta A}}{A} = \beta \Delta T$ and $\dfrac{{\Delta V}}{V} = \gamma \Delta T$. Then use the following data $\beta = 2\alpha $ and $\gamma = 3\alpha $ to reach the solution.

Complete step-by-step answer:
Thermal expansion refers to the tendency of a body to change its dimensions (length, area and volume) when the temperature is changed.
We know that more the temperature, more the kinetic energy in the molecules of the body. More kinetic energy means that the molecules will tend to be far away from each other, thus for the same number of molecules/atoms the space acquired by them will be more for higher temperatures.
There are only some materials that, for a certain temperature range, contract with an increase in temperature.
Linear expansion – It refers to an expansion in only one dimension (change in length). The change in length due to thermal expansion is equal to
\[\alpha = \dfrac{1}{L}\dfrac{{dL}}{{dT}}\]
Here, $L = $ length and $\dfrac{{dL}}{{dT}} = $ Rate of change of length (Rate of change of length per unit change in temperature).
The rate of change of linear dimension is
$\dfrac{{\Delta L}}{L} = \alpha \Delta T$
Area expansion – It refers to an expansion in two dimensions (change in area). The change in area due to thermal expansion is equal to
\[\beta = \dfrac{1}{A}\dfrac{{dA}}{{dT}}\]
Here, $A = $ Area and $\dfrac{{dA}}{{dT}} = $ Rate of change of area (Rate of change of area per unit change in temperature).
The rate of change of area is
$\dfrac{{\Delta A}}{A} = \beta \Delta T$
Volume expansion – It refers to an expansion in three dimensions (change in volume). The change in volume due to thermal expansion is equal to
\[\gamma = \dfrac{1}{V}\dfrac{{dV}}{{dT}}\]
Here, $V = $ Volume and $\dfrac{{dV}}{{dT}} = $ Rate of change of volume (Rate of change of volume per unit change in temperature).
The rate of change of volume is
$\dfrac{{\Delta V}}{V} = \gamma \Delta T$
We know $\beta = 2\alpha $ and $\gamma = 3\alpha $
So, $\alpha :\beta :\gamma = \alpha :2\alpha :3\alpha $ .
Hence, $\alpha :\beta :\gamma = 1:2:3$ .

Note: When we consider thermal expansion we are only referring to a change in the length (in case of linear expansion), area (area expansion) and volume (volume expansion). These factors change with temperature, but always remember this temperature change will not result in a phase change (solid to liquid to gas) when we are considering expansion.