Specific Heat Capacities of Gases for JEE

The specific heat capacity of any substance is defined as the amount of heat supplied per unit mass of the substance that will result in a unit rise in the temperature. It can be defined in simple terms as the amount of heat that must be added or taken away from the unit mass of a substance to change its temperature by one degree celsius. Specific heat capacity is unique for a given substance and it does not depend on the size of the system or the amount of material present in the system under consideration. This kind of property of any substance that does not depend on the size of the system is known as an ‘intensive property’.

Specific heat capacity is an intensive property of any material that describes how much energy is required to raise the temperature of the given substance. If we were to measure the heat capacity of every particle of matter, it would not be possible as there are a lot of particles even in a small sample. An intensive property is required which depends only on the type and the phase of the given substance and not on the size of the sample. This property can be applied to systems easily and this property is the specific heat. Specific heat is just the heat capacity per unit mass of the substance.

Molar Heat Capacity of a Gas

Molar heat capacity is defined as the amount of energy in the form of heat that is required to raise the temperature of one mole of any substance by one unit. The molar heat capacity depends significantly on the nature, size as well as the composition of the substance in consideration.

The dependence between the heat, heat capacity and the rise in temperature is given as,

$\Delta Q=n C \Delta T$

In another form, we can write,

$C=\dfrac{\Delta Q}{n \Delta T}$

The unit of molar heat capacity is JK-1mol-1.

In this equation,

• $\Delta Q$ is the heat supplied or the heat that is required to bring about a change of $\Delta T$ temperature in one mole of the given substance.

• n represents the amount of substance in moles. In the given case it is one.

• The constant C in the equation is known as the ‘molar heat capacity’ of the given substance.

Specific Heat Capacity of a Gas at Constant Volume

Specific heat capacity of a gas at constant volume or CV is the amount of heat absorbed or released per unit mass of the substance for a change in temperature at a constant volume. In simple words, it can be said that CV is the heat energy transferred between a system and the surroundings without any volume change. CV is the molar heat capacity when the volume is kept constant.

We can write the specific heat capacity of the gas at constant volume as,

$C_{V}=\left(\dfrac{\Delta Q}{n \Delta T}\right)_{V}$

According to the first law of thermodynamics,

$\Delta Q=\Delta U+P \Delta V$

$\Delta Q$ is the change in heat energy of the system, $\Delta U$ is the change in internal energy of the system, P is the pressure and $\Delta$ is the change in volume of the system. Since in this case, there is no change in volume or since $\Delta V$ is zero we can write $\Delta Q$ as,

$\begin{align} &\Delta Q=\Delta U+P \cdot 0 \\ &\Delta Q=\Delta U \end{align}$

This means,

$C_{V}=\left(\dfrac{\Delta U}{n \Delta T}\right)_{V}$ ….(1)

Specific Heat Capacity of a Gas at Constant Pressure

Specific heat capacity of a gas at constant pressure or CP is the amount of heat absorbed or released per unit mass of the substance for a change in temperature at a constant pressure. It can be said that CP is the heat energy transferred between a system and its surroundings at a constant pressure. CP represents the molar heat capacity of the system when the pressure is kept constant. It is to be noted that the temperature change will always cause a change in the enthalpy of the system. Enthalpy is defined as the change in heat energy of a system at constant pressure.

We can write the expression for CP as,

$C_{P}=\left(\dfrac{\Delta Q}{n \Delta T}\right)_{P}$ …. (2)

Now, since we know that the change in heat at constant pressure is enthalpy, we can write Cp in terms of enthalpy H as,

$C_{P}=\left(\dfrac{\Delta H}{n \Delta T}\right)_{P}$

Relationship Between Specific Heat Capacities of Gases

The specific heat capacities for an ideal gas are interrelated. Let us have a look at the small derivation for establishing the relationship between the Specific heat capacity at constant volume and pressure.

We know that, from the first law of thermodynamics,

$\Delta Q=\Delta U+P \Delta V$.

From equation (1) mentioned above in the article, we can write:

$\Delta U=n C_{V} \Delta T$ ….(3).

Similarly, from equation (2) we can write,

$\Delta Q=n C_{P} \Delta T$….(4).

Using equations (3) and (4) we can write the first law of thermodynamics as,

$n C_{P} \Delta T=n C_{V} \Delta T+P \Delta V$ ….(5)

We can also write using the ideal gas equation of a gas,

$P \Delta V=n R \Delta T$

Where R is the universal gas constant.

So equation (5) can be written as,

$\begin{align} &n C_{P} \Delta T=n C_{V} \Delta T+n R \Delta T \\ &n C_{P} \Delta T=n \Delta T\left(C_{V}+R\right) \\ &C_{P}=C_{V}+R \\ &C_{P}-C_{V}=R \end{align}$

The above equation is the relation between the specific heat capacities and it's also known as the “Mayers Equation”.

Conclusion

The specific heat capacity of a substance can be defined as the amount of heat that should be supplied to a system per unit mass for a unit rise in temperature. The molar heat capacity is defined as the amount of heat supplied to one mole of a substance for a unit rise in temperature. The expression for molar heat capacity is $C=\dfrac{\Delta Q}{n \Delta T}$.

The specific heat capacity at constant volume is the amount of heat supplied to a system to raise its temperature by one unit without a volume change. Its expression is $C_{V}=\left(\dfrac{\Delta U}{n \Delta T}\right)_{V}$. The specific heat capacity at constant pressure is the amount of heat supplied to a system to raise its temperature by one unit without change in its pressure. The expression is $C_{P}=\left(\dfrac{\Delta H}{n \Delta T}\right)_{P}$. The relationship between CP and CV is given as, $C_{P}-C_{V}=R$.