Specific Heat, Specific Heat at constant Volume and specific Heat at constant Pressure

Specific Heat

It is the amount of heat required to raise the temperature of unit mass of substance through unit degree. It is denoted by C.

C = Q/m×dT

Unit of C = J/Kg.K

If Q is the amount of heat transferred during any type of process where m is the mass of gas and dT = temperature difference. Then,

C = Q/m×dT

Specific Heat at constant Volume

It is the amount of heat required to raise the temperature of unit mass of substance through unit degree, when the volume is kept constant during change of state.

Let Qconst. 𝗏 is the heat transferred during constant volume process

C𝗏 = Specific heat at constant volume

m = mass

dT = change in temperature

Then C𝗏 = Qconst. 𝗏/m×dT

Let us consider a container which is non movable. Let the system is full of gases.

Let P₁ V₁ T₁ are initial properties.

Let the system is heated and by this heating the pressure and temperature will increase but the volume will remain constant.

Let P₂ V₂ T₂ are final properties.

As during this whole process volume remains constant, so this is a constant volume process. Therefore V­₁ = V₂ = V

From 1st law of thermodynamics, we know that heat supplied during any change of state can be utilized for two functions i.e to increase the internal energy and to do some work.

Therefore for this process

Q₁₋₂ = dU + W₁₋₂ -----(1)

As this is constant volume process

dU = mCvdT { Joule’s Law }

dU = Q₁₋₂ – W₁₋₂

v₂

W₁₋₂ = ʃ PdV

v₁

As there is no displacement work during constant volume process.

Therefore W₁₋₂ = 0

Therefore eq. (1) becomes

Q₁₋₂ = dU

Q₁₋₂ = mC𝗏dT -----(2)

Therefore total heat supplied during constant volume process will be utilized only to increase the internal energy of the system.

Specific Heat at constant Pressure

It is the amount of heat required to raise the temperature of unit mass of substance through unit degree, when the pressure is kept constant during change of state. It is denoted by Cp.

Let us consider a cylinder which is full of gas having properties P₁ V₁ T₁. And let a movable lid is attached at the top of the cylinder. Let the gas is heated by some external source and pressure and temperature will tend to increase.

But whenever the pressure will increase the movable lid will move upward to maintain the pressure constant and during this process the volume will increase.

Let P₂ V₂ T₂ are the final properties as the pressure is kept constant. Therefore,

P₁ = P₂ =P

From 1st law of thermodynamics

Q₁₋₂ = dU + W₁₋₂ -----(1)

Q₁₋₂ = mC𝗉dT

dU = mC𝗏dT { Joule’s Law }

v₂ v₂

W₁₋₂ = ʃ PdV = P ʃ dV

v₁ v₁

W₁₋₂ = P( V₂ – V₁)

By inserting above values in eq. (1)

Q₁₋₂ = mC𝗏dT + P( V₂ – V₁) -----(2)

Relation between specific heat at constant volume and specific heat at constant pressure

We know that heat transferred during constant pressure process

Q₁₋₂ = dU + W₁₋₂ -----(1) {from 1st law of thermodynamics}

For constant pressure process

Q₁₋₂ = mC𝗉dT

dU = mC𝗏dT

W₁₋₂ = P(V₂ – V₁)

Put these values in eq. (1)

mC𝗉dT = mC𝗏dT + P(V₂­ – V₁)

mC𝗉dT = mC𝗏dT + P₂V₂ – P₁V₁

where P₂ = P₁ = P { constant pressure process}

PV = mRT { characteristic gas eq.}

Now here P₁V₁ = mRT₁

P₂V₂ = mRT₂

mC𝗉dT = mC𝗏dT + ( mRT₂ – mRT₁ )

mC𝗉dT = mC𝗏dT + mR ( T₂ – T₁ )

C𝗉 = C𝗏 + R -----(2)

Therefore C𝗉 ˃ C𝗏

From eq. (2) it is clear that specific heat at constant pressure is always greater than specific heat at constant volume. This is due to the reason that heat supplied during constant pressure process for unit degree rise in temperature is utilized for two functions. Whereas during constant volume process all the heat supplied is used to increase the internal energy of the system.

Other relations of C𝗉 and C𝗏

C𝗉/C𝗏 = n { polytropic index}

It is the ratio of specific heat at constant pressure to the specific heat at constant volume. Its value is always greater than one.

We know that

C𝗉 – C𝗏 = R -----(1)

Dividing eq. (1) by Cv

(C𝗉 – C𝗏)/C𝗏 = R/C𝗏

(C𝗉/C𝗏) – 1 = R/C𝗏

n – 1 = R/C𝗏

R = C𝗏 ( n - 1 ) -----(2)

Now dividing eq. (1) by C𝗉

(C𝗉 – C𝗏)/C𝗉 = R/C𝗉

1 – ( C𝗏 – C𝗉 ) = R/C𝗉

1 – ( 1/n ) = R/C𝗉

R = C𝗉 ( n – 1 )/n -----(3)