Question

Dimensions of specific heat are:
A. $\left[ {M{L^2}{T^{ - 2}}K} \right]$
B. $\left[ {M{L^2}{T^{ - 2}}{K^{ - 1}}} \right]$
C. $\left[ {M{L^2}{T^2}{K^{ - 1}}} \right]$
D. $\left[ {{L^2}{T^{ - 2}}{K^{ - 1}}} \right]$

Answer 1

Recall the concept of specific heat of a material. Given the definition of the term specific heat of a material. Derive a mathematical expression for specific heat from the definition. Substitute the dimensions of all the physical quantities in the mathematical expression and determine the dimensions of the specific heat.

Complete step by step answer:
The specific heat of a material is the quantity of heat required in order to increase the temperature of unit mass of the material by a unit temperature. The mathematical expression for specific heat is given by
$C = \dfrac{Q}{{m\theta }}$ …… (1)
Here, $C$ is the specific heat of the material, $Q$ is the heat required to increase the temperature of the material, $m$ is the mass of the material and $\theta$ is the temperature of the material.

We know that the dimensions of the heat are the same as dimensions of work or energy.
Hence, the dimensional formula for heat energy is $\left[ {M{L^2}{T^{ - 2}}} \right]$. The dimensional formula for mass is $\left[ {M{L^0}{T^0}} \right]$. The dimensional formula for temperature is $\left[ {{M^0}{L^0}{T^0}K} \right]$. We can determine the dimensions of specific heat of a material or substance by substituting dimensions of all the physical quantities in the mathematical expression of specific heat.

Substitute $\left[ {M{L^2}{T^{ - 2}}} \right]$ for $Q$, $\left[ {M{L^0}{T^0}} \right]$ for $m$ and $\left[ {{M^0}{L^0}{T^0}K} \right]$ for $\theta$ in equation (1).
$C = \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {M{L^0}{T^0}} \right]\left[ {{M^0}{L^0}{T^0}K} \right]}}$
$\Rightarrow C = \left[ {M{L^2}{T^{ - 2}}} \right]\left[ {{M^{ - 1}}{L^0}{T^0}} \right]\left[ {{M^0}{L^0}{T^0}{K^{ - 1}}} \right]$
$\therefore C = \left[ {{L^2}{T^{ - 2}}{K^{ - 1}}} \right]$

Therefore, the dimensions of specific heat are $\left[ {{L^2}{T^{ - 2}}{K^{ - 1}}} \right]$. Hence, the correct option is D.

Note: The students should not get confused that the dimensions of the heat energy are taken same as that of the work or energy. The dimensions of a physical quantity can be determined from the units of the physical quantities. Since the units of all the quantities work done, heat energy and energy is the same which is joule, heat energy has the same dimensions of work and energy.