## Define Heat Capacity

The ratio of heat absorbed by a substance to the temperature change is known as heat capacity. In terms of the actual amount of material being considered, most frequently a mole, it is usually stated as calories per degree (the molecular weight in grams). Specific heat is the heat capacity in calories per gram. The calorie is defined by the specific heat of water, which is defined as one calorie per degree Celsius.

Heat capacity is the ratio of heat absorbed by a material to the temperature change. Therefore, the temperature change in a body is directly proportional to the heat transferred to the given body.

### What are Heat Capacity C, C_{p}, and C_{v}?

**Definition of molar Heat Capacity (C)**

The total amount of energy in the form of heat needed to increase the temperature of 1 mole of any substance by 1 unit is called the molar heat capacity (C) of that substance. It also significantly depends on the nature, size and composition of a substance in a system.

q = n C ∆T

Where,

q is the heat supplied or needed to bring about a change in temperature (∆T) in 1 mole of any given substance,

n is the amount in moles,

The constant C is known as the molar heat capacity of the body of the given substance.

\[C_{p}\]

In a system, C_{p} is the amount of heat energy released or absorbed by a unit mass of the substance with the change in temperature at a constant pressure. In other words, under constant pressure, it is the heat energy transfer between a system and its surroundings. So, C_{p} represents the molar heat capacity, C when pressure is constant. The change in temperature will always cause a change in the enthalpy of the system.

Enthalpy (∆H) is the heat energy absorbed or released by the system. Furthermore, enthalpy change occurs during the change of phase or state of a substance.

For example, when a solid changes to its liquid form (i.e., the change from ice to water), the enthalpy change is called the heat of fusion. When a liquid changes to its gaseous form (i.e., the change from water to water vapour), the enthalpy change is called heat vaporisation.

The system absorbs or releases heat without the change in pressure in that substance, then its specific heat at constant pressure, C_{p} can be written as:

\[C_{p} = [\frac{dH}{dT}]_{p}\] --- (1)

where Cp represents the specific heat at constant pressure; dH is the change in enthalpy; dT is the change in temperature.

**\[C_{v}\]**

During a small change in the temperature of a substance, C_{v} is the amount of heat energy absorbed/released per unit mass of a substance where volume does not change. In other words, C_{v} is the heat energy transfer between a system and its surroundings without any change in the volume of that system. C_{v} represents the molar heat capacity C when the volume is constant. Under a constant volume, the volume of a substance does not change, so the change in volume is zero.

As the term is related to the internal energy of a system, which is the total of both potential energy and kinetic energy of that system. The system absorbs or releases heat without change in volume of that substance, then its specific heat at constant volume, C_{v} can be:

\[C_{v} = [\frac{dU}{dT}]_{v}\] --- (2)

Where,

C

_{v}represents the specific heat at constant volume;dU is the small change in the internal energy of the system;

dT is the change in temperature of the system.

Relationship Between C_{p} and C_{v}

According to the first law of thermodynamics:

\[\Delta Q = \Delta U + \Delta W\] where, \[\Delta Q\] is the amount of heat that is given to the system, U is the change in internal energy and W is the work done.

We can write,

\[\Delta Q = \Delta U + P\Delta V as \Delta W = P\Delta V\]

Since \[\Delta Q = \Delta n C_{p}\Delta T \, and \, \Delta U = \Delta n_{v}C \Delta T\]

Therefore,

\[n C_{p} \Delta T = n C_{v} \Delta T + P\Delta V\] ----- (3)

As we that PV = nRT

At T1 Kelvin:\[PV_{1} = nRT_{1}\] ------------ (a)

At T2 Kelvin: \[PV_{2} = nRT_{2}\] ------------ (b)

Subtracting the equation (a) from (b):

\[PV_{2} - PV_{1} = nRT_{2} - nRT_{1}\]

\[P(V_{2} - V_{1}) = nR (T_{2} - T_{1})\]

Where,

\[V_{2} - V_{1} = \Delta V\] and \[T_{2} - T_{1} = \Delta T\]

Therefore,

\[P \Delta V = nR \Delta T\]

Substituting the value of \[P \Delta V\] in equation (3), we get,

\[n C_{p} \Delta T = n C_{v} \Delta T + nR \Delta T\]

\[n C_{p}\Delta T = n\Delta T (C_{v} + R)\]

\[C_{p} = C_{v} + R\]

Or

\[C_{p} - C_{v} = R\]

The following relationship can be given considering the ideal gas behaviour of a gas.

\[C_{p} - C_{v} = R\]

Where, R is called the universal gas constant.

### Heat Capacity Ratio

In thermodynamics, the heat capacity ratio or ratio of specific heat capacities (C_{p}:C_{v}) is also known as the adiabatic index. It is the ratio of two specific heat capacities, C_{p} and C_{v} is given by:

The Heat Capacity at Constant Pressure (C_{p})/ Heat capacity at Constant Volume(C_{v})

The isentropic expansion factor is another name for heat capacity ratio that is also denoted for an ideal gas by γ (gamma). Therefore, the ratio between C_{p} and C_{v} is the specific heat ratio, γ.

So,

\[\gamma = \frac{C_{p}}{C_{v}}\]

It is essential to study the heat capacity ratio for applying in the reversible processes of thermodynamics, especially where ideal gases are involved.

For Example, to study the relation with degrees of freedom,

The heat capacity ratio (gamma, γ) for an ideal gas can be related to the degrees of freedom ( f ) of gas molecules by the formula:

\[\gamma = 1 + 2f\] or\[f = 2\gamma - 1\]

The specific heat of gas at constant volume in terms of degree of freedom 'f' is given as:

\[C_{v} = (\frac{f}{2})R\]

Also,

\[C_{p} - C_{v} = R\]

Therefore,

\[C_{p} = (\frac{f}{2})R + R = R(1 + \frac{f}{2})\]

Now, ratio of specific heats γ is given as:

\[\gamma = \frac{C_{p}}{C_{v}} = \frac{R(1 + \frac{f}{2})}{(\frac{f}{2})R}\]

\[= \frac{(2 + f)}{f}\]

Or \[\gamma = \frac{C_{p}}{C_{v}}\]

So, we can also say that,\[\frac{C_{p}}{C_{v}} = (1+\frac{2}{f})\], where, f is the degree of freedom.

Monoatomic gas has only one translational motion, hence three translational degrees of freedom.

The average energy of the molecule E, at temperature T, is given by:

\[E = \frac{1}{2}mv_{x}^{2} + \frac{1}{2}mv_{y}^{2} + \frac{1}{2}mv_{z}^{2}\]

where, m is the mass of the molecule, v (\[V_{x}\],\[V_{y}\] and \[V_{z}\]) is the momentum of a molecule along with x-axis, y-axis and z-axis.

The Law of Equipartition gives,

\[E = \frac{1}{2}k_{B} T + \frac{1}{2}k_{B} T + \frac{1}{2}k_{B} T\]

Here, \[K_{B}\] is average translational kinetic energy and T is the temperature.

Energy per molecule of gas is given by,

\[U = \frac{3}{2}k_{B}T\]

\[= \frac{3}{2} RT\]

Thus,\[\frac{dU}{dT} = R\]

But, \[\frac{dU}{dT} = C_{v}\]

Thus,\[C_{v} = \frac{3}{2} R\]

Now, \[C_{p} - C_{v} = R\]

Then, \[C_{p} = R + C_{v}\]

This gives,\[\gamma = \frac{C_{p}}{C_{v}} = \frac{{\frac{5}{2}R}}{{\frac{3}{2}R}} =\frac{5}{3} = .67\]

For a diatomic gas (such as, H_{2}, O_{2} and N_{2}), it has 5 as degrees of freedom (3 as translational and 2 as rotational degrees of freedom at room temperature; whereas, except at high temperatures, the vibrational degree of freedom is not involved).

### Why is C_{p} Greater than C_{v}?

The values indicated by C_{p} and C_{v} are the specific heats of an ideal gas. These indicate the quantity of heat that can increase the temperature of unit mass by 1°C.

By the first law of thermodynamics,

\[\Delta Q = \Delta U + \Delta W\]

Where,

\[\Delta Q\] is the amount of heat that is given to the system, \[\Delta U\] is the change in internal energy and W is the work done.

So, at constant pressure, heat is absorbed not only for increasing the internal energy (function of temperature) but for doing work as well. Whereas, at constant volume, heat is absorbed only for raising the internal energy and not for doing any kind of work on the system as (for a closed system):\[W = P \Delta V\], where W is the work done. Here, \[\Delta V = 0\]. (A closed system is also one of the crucial conditions for constant volume). Hence, the specific heat at constant pressure is more than the specific heat at a constant volume, i.e., \[C_{p} > C_{v}\].