Question

Consider the following statements
1. The coefficient of linear expansion has dimension ${K^{ - 1}}$.
2. The coefficient of volume expansion has dimension ${K^{ - 1}}$.
Which of the above is/are correct:
A) Both 1 and 2 are correct
B) 1 is correct but 2 is wrong
C) 2 is correct but 1 is wrong
D) Both 1 and 2 are wrong.

Answer 1

Hint: Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. Atoms and molecules in a solid, for instance, constantly oscillate around its equilibrium point. This kind of excitation is called thermal motion. When a substance is heated, its constituent particles begin moving more, thus maintaining a greater average separation with their neighboring particles. The degree of expansion divided by the change in temperature is called the material’s coefficient of thermal expansion. It generally varies with temperature.
A dimension is a measure of a physical variable without numerical values. Use the formula for the coefficient of linear expansion ${\alpha _L} = \dfrac{{\Delta L}}{{L\Delta T}}$ and the coefficient of volume expansion ${\alpha _V} = \dfrac{{\Delta V}}{{V\Delta T}}$.

Complete solution:
Express the formula of the coefficient of the linear expansion of a matter
\[\therefore {\alpha _L} = \dfrac{{\Delta L}}{{L\Delta T}}\] , where $\Delta L$ the change in the length is, $L$ is the initial length, and $\Delta T$ is the change in the temperature.
Dimension of length is \[{L^1}\] , and the dimension of temperature is ${K^1}$ . Therefore
The dimension of the coefficient of linear expansion $ = \dfrac{{{L^1}}}{{{L^1}{T^1}}}$
$ \Rightarrow $ The dimension of \[{\alpha _L} = \dfrac{1}{{{K^1}}} = {K^{ - 1}}\]
Similarly express the formula for the coefficient of volume expansion of a matter
$\therefore {\alpha _V} = \dfrac{{\Delta V}}{{V\Delta T}}$ , where $\Delta V$ is the change in the volume, $V$ is the initial volume of the, and $\Delta T$ is the change in temperature.
We know the dimension of the volume is ${L^3}$ and the dimension of the temperature is ${T^1}$ . Therefore,
The dimension of the coefficient of the volume expansion $ = \dfrac{{{L^3}}}{{{L^3}K}}$
$ \Rightarrow $ The dimension of ${\alpha _V} = \dfrac{1}{{{K^1}}} = {K^{ - 1}}$

Hence the option A is correct.

Note: Unit and dimension may be confusing. Dimensions are physical quantities that can be measured, whereas units are arbitrary names that correlate to particular dimensions to make it relative. All units for the same dimension are related to each other through a conversion factor. For example, 1 m is equal to 100 cm. Here units are different but the dimension is the same.