# All About Heat Capacity and How it Relates to Cp and Cv

What is the definition of Heat Capacity?

Let’s begin by answering what is heat capacity? The quantifiable physical quantity that describes the volume of heat necessary to change a substance’s temperature by a certain amount is heat capacity (typically indicated by a capital C, frequently with subscripts).

Heat capacity is measured in joules per kelvin (J/K) in SI units.

The ratio of the quantity of heat energy delivered to an item to the ensuing increase in temperature is defined as the object’s heat capacity (symbol C).

The heat q necessary to bring about a ∆T variation in temperature of one mole of any stuff is represented by the formula q = n C ∆T. The molar heat capacity of the body is represented by the constant C.

As a consequence, every substance’s molar heat capacity is defined as the amount of heat energy necessary to raise the temperature of one mole of that substance by one unit. It is dependent on the system’s type, size, and composition.

The following is what this formula represents:

• q is the amount of heat that is provided or required to cause a change in temperature (∆T) in one mole of any given material.
• n is the number of moles.
• The molar heat capacity of the body of a given substance is known as the constant C.

Cp: Cp is the quantity of heat energy emitted or absorbed by a unit mass of a substance when the temperature changes at a constant pressure in a system. In other words, it is the thermal energy transfer between a system and its surroundings under continuous pressure.

So, while pressure is constant, Cp represents the molar heat capacity, and C represents the molar heat capacity. A change in temperature will always result in a change in the system’s enthalpy.

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The heat energy received or emitted by the system is called enthalpy (∆H). Furthermore, enthalpy change happens when a substance’s phase or condition changes.

The heat of fusion, for example, is the enthalpy shift that occurs when a solid transforms into a liquid (for example, when ice transforms into water). The heat of vaporisation is the enthalpy shift that occurs when a liquid changes to a gaseous state (i.e., when water changes to water vapour).

When a system absorbs or releases heat in a material without changing its pressure, its specific heat at constant pressure, Cp, may be stated as:

pp

= \[dHdT\]\[dHdT\]

pp

———————–(1)

where Cp denotes specific heat at constant pressure, dH denotes enthalpy change, and dT denotes temperature change.

Cv is the amount of heat energy absorbed/released per unit mass of a substance after a slight change in temperature while volume does not vary. In other words, Cv is the thermal energy transfer between a system and its surroundings that occurs without a change in the system’s volume.

When volume is constant, Cv indicates the molar heat capacity C. The volume of a material does not vary when it has a constant volume, hence the volume change is zero.

As the name implies, internal energy refers to a system’s overall energy, which includes both potential and kinetic energy. When a system absorbs or releases heat without changing the volume of a material, the specific heat at constant volume, Cv, may be calculated as follows:

vv

= \[dUdT\]\[dUdT\]

vv

—————————(2)

where Cv denotes specific heat at constant volume, dU denotes a minor change in the system’s internal energy, and dT denotes a change in the system’s temperature.

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For an Ideal Gas, the relationship between CP and CV

From the equation q = n C ∆T, we can say:

We have at constant pressure P

qP = n CP∆T

This value is equal to the change in enthalpy, that is,

qP = n CP∆T = ∆H

Similarly, at constant volume V, we have

qV = n CV∆T

This value is equal to the change in internal energy, that is,

qV = n CV∆T = ∆U

We know that for one mole (n=1) of an ideal gas,

∆H = ∆U ∆(pV ) = ∆U ∆(RT) = ∆U R ∆T

Therefore, ∆H = ∆U R ∆T

Substituting the values of ∆H and ∆U from above in the former equation,

CP∆T = CV∆T R ∆T

CP = CV  R

CP – CV = R

Why is Cp more important than Cv?

The specific temperatures of an ideal gas are represented by Cp and Cv. This is the amount of heat required to raise the temperature of a unit mass by 1°C.

Thermodynamics’ first law states that

ΔQ = ΔU ΔW

where ΔQ denotes the quantity of heat supplied to the system, ΔU denotes the change in internal energy, and ΔW is the work performed.

As a consequence, heat is absorbed under constant pressure not just to increase internal energy (a function of temperature), but also to perform work. At constant volume, heat is absorbed just to raise internal energy, not to do any work on the system, as follows (for a closed system): W = PΔV, where W is the work done.

Here, ∆V = 0. (A closed system is also one of the most important requirements for maintaining a consistent volume.) As a result, the specific heat at constant pressure is larger than at constant volume, resulting in Cp > Cv.