Heat Capacity: Cp and Cv Relation, Types & SI Unit

Heat Capacity or Thermal Capacity, in Thermodynamics, is defined as the amount of heat required to change the temperature of an object by one unit. Being an inherent property of a substance, heat capacity can also be defined as the ratio of the amount of heat absorbed by a system to the change in temperature.

Read More: Concepts in Chemistry

Key Terms: Heat Capacity, Specific Heat Capacity, Cp, Cv, Molar Heat Capacity, Isobaric process, isochoric process. 

---

What is Heat Capacity?

[Click Here for Sample Questions]

Heat is the transfer of energy from one body to another owing to the temperature difference between them. In a thermodynamic system, each molecule or compound reacts differently to heat. The temperature of a body rises when it absorbs heat or vice versa. When a body absorbs heat, it is converted into kinetic energy of the particles, resulting in the rise of temperature. As a result, the temperature change is directly proportional to the amount of heat transfer. 

Heat Capacity of Water Graph

Heat Capacity is calculated by the formula:

Q = C\(\Delta\)T

so, 

C= Q/\(\Delta\)T

where, 

Q is the amount of heat required in Joules

\(\Delta\)T is the change in temperature (Final temperature – Initial temperature) in K

C is the heat capacity 

Therefore the S.I unit of heat capacity is joule per kelvin (J/K). 

Heat Capacity can also be calculated using specific heat capacity (c) by the mathematical equation:

Q = mc\(\Delta\)T

Where,

Q is the heat capacity in Joules

m is the mass in grams

c is the specific heat of an object in J/g °C or K

\(\Delta\)T is the change in the temperature in °C or K

Also Read:

---

Types of Heat Capacity

[Click Here for Sample Questions]

There are three types of heat capacity namely, 

• Molar Heat Capacity (C), 
• Specific Heat Capacity,
• Cp and Cv.

Molar Heat Capacity (c_m)

Molar heat capacity of any substance is the amount of heat energy required to raise the temperature of one mole of a substance by one unit.

Molar heat capacity is calculated as: 

c_m = C/n

c_m = Q/\(\Delta\)T/n

c_m = Q/n\(\Delta\)T

where, 

Q is the amount of heat required to change the temperature (\(\Delta\)T) in one mole of any given substance. 

C the molar heat capacity of the body of the given substance.

n is the amount in moles 

Unit of Molar heat capacity is given by,

Q = Joules (J)

n = moles

\(\Delta\)T = K

Thereby substituting in the equation above, we get cm = (J/(K⋅mol), or J K⁻¹ mol⁻¹.

Specific Heat Capacity (c)

Specific heat capacity of any substance is defined as “the amount of heat required to change the temperature of a unit mass of the substance by 1 degree.” 

The below-mentioned formula can be used to calculate specific heat capacity values:

Q = mc\(\Delta\)T

Therefore specific heat capacity (c) = Q/(m\(\Delta\)T)

The SI unit of specific heat capacity is joule per kelvin per kilogram, J/(kg . K) 

Specific Heat Capacity Video Explanation

Constant Pressure (C_p)

At constant pressure, the amount of heat energy released or absorbed by a unit mass of substance with a change in temperature is known as molar heat capacity at constant pressure or Cp.

At constant pressure, \(\delta\)Q = dU + PdV (isobaric process)

Cp can be written as: 

Cp = [dH/dT]p

where

Cp represents the specific heat at constant pressure

dH is the change in enthalpy

dT is the change in temperature

Constant Volume (C_v)

Cv or the molar heat capacity at constant volume is the amount of heat energy released/absorbed per unit mass of a substance at constant volume during a small change in the temperature of a substance. 

At constant volume, dV = 0, \(\delta\)Q = dU (isochoric process)

Cv can be written as:

C_v = [dU/dT]_v

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. 

Also Read:

---

Relationship between C_P and C_V

[Click Here for Sample Questions]

According to the first law of thermodynamics,

 q = n cm \(\Delta\)T. . . . . . . . . . (1) 

At constant pressure P, equation one changes to

qP = n CP\(\Delta\)T

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

qP = n CP\(\Delta\)T = \(\Delta\)H. . . . . . . . . . (2)

Similarly, at constant volume V, we have

qV = n CV\(\Delta\)T

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

qV = n CV\(\Delta\)T = \(\Delta\)U. . . . . . . . . . . (3)

 The formula for one mole (n=1) of an ideal gas,

\(\Delta\)H =\(\Delta\)U + \(\Delta\)(PV )

\(\Delta\)H = \(\Delta\)U +\(\Delta\)(RT)

On rearranging the above equation, we have

ΔH = ΔU + R ΔT

Therefore, ΔH = ΔU + R ΔT

Substituting the values of ΔH and ΔU from the equations (2) and (3) in the former equation (1), 

nCPΔT = nCVΔT + R ΔT

For (n=1), the above equation can be written as:

CPΔT = CVΔT + R ΔT

By taking ΔT as a common term, then

 

C_p×ΔT=(C_v+R)ΔT

By cancelling the terms \(\Delta\)T on both sides, then

C_P = C_V + R
C_P – C_V = R

Read More: Formula Related to Heat Capacity

---

Things to Remember

[Click Here for Sample Questions]

• The SI unit of molar heat capacity is joule per kelvin per mole (J/(K⋅mol), or J K⁻¹ mol⁻¹
• The calorimeter is the device used for measuring heat.
• The molar heat capacity at constant pressure CP is always greater than the molar heat capacity at constant volume Cv because the substance expands when heat is supplied at constant pressure. 
• When the size of the body increases with temperature (heat), it results in thermal expansion of that body. 

Also Read: