Question

Define heat capacity and state its SI unit.

Answer 1

Hint: Thermal capacity, also known as heat capacity is defined as the amount of heat required to raise the temperature of the whole body (mass m) through ${{1}^{0}}$C or 1 K. It is given as $C=\dfrac{Q}{\Delta T}$. With the help of this formula, calculate the units of heat capacity.

Complete step by step answer:
Thermal capacity, also known as heat capacity, is defined as the amount of heat required to raise the temperature of the whole body (mass m) through ${{1}^{0}}$C or 1 K.
It is represented by C.
When some of heat is supplied to a body of mass m, its temperature rises. Let the change in temperature of the body be $\Delta $T and the heat energy supplied be Q. Then the relation between the heat capacity (C) of the body, the change in temperature ($\Delta $T) of the body and the supplied heat Q is given as $Q=C\Delta T$.
Therefore, $C=\dfrac{Q}{\Delta T}$.
The value of thermal capacity of a body depends upon the nature of the body and its mass.
Let us calculate the units of heat capacity.
First, let us calculate the dimensional formula of heat capacity.
The dimensional formula of heat capacity will be $\left[ C \right]=\dfrac{\left[ Q \right]}{\left[ \Delta T \right]}$ …… (i).
The dimensional formula of temperature is given as $\theta $. Therefore, the dimensional formula of change in temperature ($\Delta $T) will be $\theta $.
Heat is a form of energy and we know that the dimensions of work done on a body and energy are the same. We also know that work done is equal to force times displacement,
i.e. W = Fd.
Therefore, dimensional formula of heat is $\left[ Q \right]=\left[ W \right]=\left[ Fd \right]=\left[ F \right]\left[ d \right]$.
The dimension formula of force is $\left[ F \right]=\left[ ML{{T}^{-2}} \right]$.
The dimensional formula of displacement is [L].
Hence, $\left[ Q \right]=\left[ F \right]\left[ d \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
Substitute the dimensional formulas of heat and temperature change in equation (i).
Therefore, we get
$\left[ C \right]=\dfrac{\left[ Q \right]}{\left[ \Delta T \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ \theta \right]}=\left[ M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}} \right]$.
With this dimensional formula we get the unit of heat capacity in MKS system as $kg{{m}^{2}}{{s}^{-2}}{{K}^{-1}}$.
The SI unit of heat is Joule (J) and temperature is kelvin (K).
Hence, the SI unit of heat capacity is $\dfrac{J}{K}$.

Note: Do not confuse between heat capacity and specific heat capacity of a body. Specific heat capacity of a body is the amount of heat required to raise the temperature of the unit mass of the body by ${{1}^{0}}$C or 1 K. Heat capacity depends on the property of the material and mass of the body whereas specific heat capacity only depends on the property of the material.