- #1
Dr. Who
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Hi All,
I have a little query concerning the derivation of PV γ = constant. In my textbook of Physics, first they give the equation for adiabatic process using the first law of Thermodynamics, as;
where,
ΔEint ⇒ change in internal energy and W ⇒ workdone
Then, they used the relation:
Where,
Q ⇒ heat
'n' ⇒ no. of moles
Cv ⇒ Molar Heat capacity at constant volume
dT ⇒ Change in temperature
Now, for an isochoric process;
Substituting dEint from eq.(2) into eq.(1)
As thermodynamic work is given as W = -PdV
Now, writing equation of state of the gas in differential form as;
Using eq.(3)
Using the relation Cp = Cv + R into the above equation
Now, dividing eq.(4) by eq.(3)
where, γ ⇒ ratio of molar heat capacities
Rearranging the above equation:
Integrating both sides with the initial state 'i' and final state 'f' being the lower and upper limits respectively, gives;
which can be written as: PV γ = constant
Now, my query was that,
1. why have they substituted the internal energy from an isochoric process into an adiabatic process (Substituting dEint from eq.(2) into eq.(1))? Considering this substitution, can we say that heat absorbed in the isochoric process is equivalent to the work done in an adiabatic process?
2. What is the physical significance of γ on a curve of PV γ = constant ? (Please do not go into the details of poltropic processes)
I have a little query concerning the derivation of PV γ = constant. In my textbook of Physics, first they give the equation for adiabatic process using the first law of Thermodynamics, as;
dEint = W → (1)
where,
ΔEint ⇒ change in internal energy and W ⇒ workdone
Then, they used the relation:
Cv = Q / ndt
Q ⇒ heat
'n' ⇒ no. of moles
Cv ⇒ Molar Heat capacity at constant volume
dT ⇒ Change in temperature
Now, for an isochoric process;
Q = dEint
∴ dEint = nCvdT → (2)Substituting dEint from eq.(2) into eq.(1)
⇒ W = nCvdT
As thermodynamic work is given as W = -PdV
∴ -PdV = nCvdT
⇒ PdV = - nCvdT → (3)
⇒ PdV = - nCvdT → (3)
Now, writing equation of state of the gas in differential form as;
d(PV) = d(nRT)
⇒ PdV + VdP = nRdT⇒ - nCvdT + VdP = nRdT
⇒ VdP = nCvdT + nRdT
⇒ VdP = nCvdT + nRdT
Using the relation Cp = Cv + R into the above equation
⇒ VdP = nCpdT → (4)
Now, dividing eq.(4) by eq.(3)
⇒ VdP / PdV = -Cp / Cv
or VdP / PdV = -γ
or VdP / PdV = -γ
Rearranging the above equation:
dP / P = -γ dV / V
∫ dP / P = -γ ∫ dV / V
⇒ PiViγ = PfVfγ
which can be written as: PV γ = constant
Now, my query was that,
1. why have they substituted the internal energy from an isochoric process into an adiabatic process (Substituting dEint from eq.(2) into eq.(1))? Considering this substitution, can we say that heat absorbed in the isochoric process is equivalent to the work done in an adiabatic process?
2. What is the physical significance of γ on a curve of PV γ = constant ? (Please do not go into the details of poltropic processes)