# Heat Capacity: Types and Factors Affecting Heat Capacity

## Heat Capacity

The amount of heat required to change its temperature by one degree of a substance is called heat capacity.

If 'Q' calories of heat absorbed by mass 'm' and the temperature raises from T_{1} to T_{2} then the heat capacity 'C' is given by the expression-

C = Q/m(T_{2} − T_{1})

or, C = Q/ΔT

or, Q = CΔT

## Molar Heat Capacity

The amount of heat required to change its temperature by one degree of one mole of a substance. Since the heat capacity(C) varies with temperature, the true value will be-

C = dQ/dT

where, dQ is a small quantity of heat absorbed by the system producing a small temperature raise dT. Thus the molar heat capacity may be defined as the ratio of the amount of heat absorbed to the rise in temperature.

## Unit of Heat Capacity

Unit of molar heat capacity are-

calories per degree per mole or in SI unit- joules per degree per mole.

Since heat capacity is a path function so it is necessary to specify the process by which the temperature raised by one degree. There are two processes for this one is molar heat capacity at constant pressure and other is molar heat capacity at constant volume.

## Molar Heat Capacity at Constant Pressure (C_{p})

The amount of heat required to change its temperature by one degree of a substance at constant pressure. It is denoted by C_{p}.

So, C_{p} = dQ_{p}/dT ----- Eq:1

we know that, H = E + PV

or, dH = dE + PdV + VdP

At constant pressure, dP = 0

So the above equation becomes-

dH = dE + PdV

or, dH = dE + W (as = PdV)

or, dH = Q_{p} (as Q_{P} = dE + PdV)

Putting the value of Q_{p} in Eq:1, we get-

C_{P} = dH_{P}/dT

or, dH = C_{P} dT

on integrating this equation, we get-

or, H_{2} − H_{1} = C_{P} (T_{2} − T_{1})

or, ΔH = C_{P} (T_{2} − T_{1})

This equation shows that the dependency of enthalpy on temperature.

## Molar Heat Capacity at Constant Volume (C_{v})

The amount of heat required to change its temperature by one degree of a substance at constant volume. It is denoted by C_{V}.

So, C_{V} = dQ_{v}/dT ----- Eq:2

From first law of thermodynamics, we know that-

Q_{V} = dE + W = dE + PdV

At constant V, dV = 0

so, the above equation becomes-

Q_{V} = dE

Putting the value of Q_{v} in Eq:2, we get-

C_{v} = dE/dT

or, dE = C_{v}dT

on integrating this equation, we get-

or, E_{2} − E_{1} = C_{v} (T_{2} − T_{1})

or, ΔE = C_{V} (T_{2} − T_{1})

This equation shows that the dependency of Internal energy on temperature at constant volume.

## Relation between C_{P} and C_{V}

We know that

H = E + PV

or, H = E + RT (as PV = RT for one mole)

differentiating the above equation w.r.t T, we get-

dH/dT = dE/dT + R(dT/dT)

or, C_{P} = C_{V} + R

or, C_{P} − C_{V} = R

## Factors Affecting Heat Capacity

Heat capacity of a substance depends on several factors. Some of them are discussed below-

**Nature of the substance:** Different substances have different heat capacities due to their molecular structure and intermolecular forces.

**Mass:** Heat capacity is directly proportional to the mass of the substance.

**Temperature:** Heat capacity of a substance can vary with temperature.