Question

The dimensions of heat capacity are:
A. $[M{L}^{-2}{T}^{-2}{K}^{-1}]$
B. $[M{L}^2{T}^{-2}{K}^{-1}]$
C. $[M^{-1}{L}^2{T}^{-2}{K}^{-1}]$
D. $[ML{T}^{-2}K]$

Answer 1

Hint: We know that heat capacity is defined as the heat required to raise the temperature of a substance by 1 unit of temperature. Now, to reduce a quantity to its dimensional form means expressing the units of the quantity in terms of just powers of mass, length, time and temperature. In other words, reduce the units of heat capacity from the SI units to the base units by taking the heat energy and the change in temperature in terms of the base units. Following this, assign the powers of the base units to the fundamental quantities of the dimensional formula to arrive at the appropriate dimensions for heat capacity.
Formula used:
Heat capacity $C_H={\displaystyle \frac{Q}{\mathrm{\Delta }T}}$

Complete answer:
Let us begin by establishing an understanding of heat capacity.
Heat capacity, also known as thermal capacity is a measurable physical quantity that characterizes the amount of heat required to raise the temperature of the substance by $1^{\circ }C$, and is given as the ratio of the amount of heat energy Q transferred to a substance to the resulting increase in the temperature ($\mathrm{\Delta }T$) of the substance, i.e.,
$C_H={\displaystyle \frac{Q}{\mathrm{\Delta }T}}$ and is usually measured in $J^{\circ }{C}^{-1}$
Now, a dimensional formula for any physical quantity looks like $\left[M^x{L}^y{T}^z{K}^a\right]$, where M, L, T and K, are the mass, length, time and temperature respectively, and x, y, z and a are any integers representing dimensions.
Let us now try to dimensionally analyse what our definition of heat capacity looks like.
For this, let us deconstruct the heat capacity expression to its dimensional notation by evaluating the dimensions of heat transfer and temperature.
The heat energy Q transferred is measured in joules (J), which in base units is expressed as $kg.m^2.s^{-2}$
Therefore, the dimensional formula for Q will be $\left[M{L}^2{T}^{-2}\right]$
The change in temperature $\mathrm{\Delta }T$ is measured in kelvin (K), therefore, the dimensional formula for $T$ will be $\left[K\right]$
Putting this all together is the expression for heat capacity, we can get its dimensional formula as:
$\left[C_H\right]={\displaystyle \frac{\left[Q\right]}{\left[\mathrm{\Delta }T\right]}}={\displaystyle \frac{\left[M{L}^2{T}^{-2}\right]}{\left[K\right]}}=\left[M{L}^2{T}^{-2}{K}^{-1}\right]$

So, the correct answer is “Option B”.

Note:
The dimensional formula allows us to express the unit of a physical quantity in terms of the fundamental quantities mass (M), length (L), time (T) and temperature (K).
However, while reducing the dimensional formula from the base units be careful in assigning the right power to the right fundamental quantity each time as any discrepancy in the dimension will result in an anomalous nature of the quantities involved.