Specific heat of a gas
Introduction: It is very clear from previous topic that equal amount of energy is distributed among each degree of freedom according to equipartition of energy. We can make use of equipartition principle to calculate specific heat of a gas with the help of mayer's equation. This would give us a mathematical approach for calculating molar specific heat of a gas at constant volume (Cv)and constant pressure (Cp).
Monoatomic gas: Consider 1 mole of gas at enclosed in a container at constant temperature. monoatomic molecules have 3 degree of freedom. Thus average kinetic energy per atom 3/2 Kb T. Thus total internal energy per mole is given by,
U = 3/2 NA Kb T
U = 3/2 R T ----( NA Kb = R)
we know that
Cv = U/ dT
Cv = 3/2 R T / dT
Cv = 3/2 R -----(1)
From Mayer's equation
R = ( Cp - Cv )
Cp = R + Cv )
Cp = R + 3/2 R)
Cp = 5/2 R -----(2)
from (1) and (2)
γ = Cp / Cv
γ = 5/3
Diatomic gas: Consider 1 mole of gas at enclosed in a container at constant temperature. diatomic molecules have 5 degree of freedom. Thus average kinetic energy per atom 5/2 Kb T. Thus total internal energy per mole is given by,
U = 5/2 NA KbT
U = 5/2 RT ----( NA Kb = R)
we know that
Cv = U/ dT
Cv = 5/2 R T / dT
Cv = 5/2 R -----(1)
From Mayer's equation
R = ( Cp - Cv )
Cp = R + Cv )
Cp = R + 5/2 R)
Cp = 7/2 R -----(2)
from (1) and (2)
γ = Cp / Cv
γ = 7/5
* If diatomic molecule is not rigid it has additional vibration mode
U = 7/2 R T ----( NA Kb= R)
we know that
Cv = U/ dT
Cv = 7/2 R T / dT
Cv = 7/2 R -----(1)
From Mayer's equation
R = ( Cp - Cv )
Cp = R + Cv )
Cp = R + 7/2 R)
Cp = 9/2 R -----(2)
from (1) and (2)
γ = Cp / Cv
γ = 9/7
Polyatomic Gases : Polyatomic gases have 3 translational degree of freedom , 3 rotational degree of freedom and f number of vibrational degree of freedom.
By law of equipartition of energy each mole of gas has,
U = 3 R T + f RT ----( NA Kb = R)
we know that
Cv = U/ dT
Cv = 3 R T + f RT / dT
Cv = (3 + f )RT / dT
Cv = (3+ f) R -----(1)
From Mayer's equation
R = ( Cp - Cv )
Cp = R + (3+ f) R )
Cp = R + 3R + fR)
Cp = 4R + fR
Cp = (4+ f) R
from (1) and (2)
γ = Cp / Cv
γ = (4+ f) / (3+ f)
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