- The heat capacity of a substance is defined as the amount of heat required to rise of the temperature by one degree. Heat capacity per gram of substance is called specific heat and per mole called molar heat capacity.

Thus, Molar heat capacity = Molar mass × Specific heat

Cp = M × c

Cp = M × c

_{p}, Cv = M × c_{v}- Where Cp and Cv are the molar heat capacities at constant pressure and constant volume respectively. c

_{p}and c

_{v}are their specific heats.

- The specific heat at constant pressure and constant volumes are 0.125 and 0.075 Cal gm⁻¹K⁻¹ respectively. Calculate the molecular weight and the atomicity of the gas. Name the gas if possible.

- M = 40 and ⋎ = 1.66(mono-atomic), Ar(Argon). For gases, there are two heat capacities at constant volume and constant pressure. These are Represented as,

Cv = (dq/dT)v = (dU/dT)v

Cp = (dq/dT)p = (dU/dT)p + P(dV/dT)p

#### Heat capacities of an Ideal Gas

If the gas is assumed to be ideal, then,

PV = nRT and (dU/dT)v = (dU/dT)p

Again, P(dV/dT)P = nR

Thus for an ideal gas,

PV = nRT

or, P(dV/dT)p = nR

Again, Cp = (dU/dT)P + P(dV/dT)P

or, Cp = Cv + {P×(nR/P)}

or, Cp = Cv +nR

For 1 mole ideal gas

Cp = Cv +R

PV = nRT and (dU/dT)v = (dU/dT)p

Again, P(dV/dT)P = nR

Thus for an ideal gas,

PV = nRT

or, P(dV/dT)p = nR

Again, Cp = (dU/dT)P + P(dV/dT)P

or, Cp = Cv + {P×(nR/P)}

or, Cp = Cv +nR

For 1 mole ideal gas

Cp = Cv +R

Molar heat capacity |

- From the above two descriptions, it is clear that Cp and Cv. Since for Cp, some mechanical work is required as additional energy to absorbed for definite piston from volume V₁ to V₂.

Cp - Cv = Mechanical work

or,PdV = P(V₂-V₁) = PV₂-PV₁

= R(T+1) - RT

= R

or,PdV = P(V₂-V₁) = PV₂-PV₁

= R(T+1) - RT

= R

- Now let us find the expression of Cv from the point of view of Kinetic theory.

- Cv = Energy required to increase the translational kinetic energy of 1-mole gas for a rise of 1° temperature + energy required to increase the intermolecular energy of 1-mole gas for the rising temperature of 1°.

Increase of transnational K.E. = (3/2)R(T+1) - (3/2)R

= (3/2) R

For 1-mole gas for 1°C rise in temperature.

= (3/2) R

For 1-mole gas for 1°C rise in temperature.

#### Mono-atomic gases

Cv = (dU/dT)v = (3/2)

Cp = (5/2)R.

Thus Î³ = Cp/Cv = 5/3 ≃ 1.667

Cp = (5/2)R.

Thus Î³ = Cp/Cv = 5/3 ≃ 1.667

#### Poly-atomic gases

For linear Cv = (dU/dT)v = (3/2)R + R +(3N - 5)R

For non-linear Cv = (dU/dT)v = (3/2)R + (3/2)R +(3N - 6)R

where N is the number of particles.

For non-linear Cv = (dU/dT)v = (3/2)R + (3/2)R +(3N - 6)R

where N is the number of particles.

#### Molar heat capacities for diatomic molecules

For diatomic molecule N = 2.

Thus Cv = (3/2)R + R + R = (7/2)R

Cp = (9/2)R.

∴ ⋎ = Cp/Cv = 9/7 ≃ 1.286

Thus Cv = (3/2)R + R + R = (7/2)R

Cp = (9/2)R.

∴ ⋎ = Cp/Cv = 9/7 ≃ 1.286

#### Molar heat capacities for tri-atomic molecules

For triatomic molecule N = 3