Undergrad Relation between 'P' and 'V' for an Adiabatic process

[Dr. Who]

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;

dE_int = W → (1)​

where,
ΔE_int ⇒ change in internal energy and W ⇒ workdone
Then, they used the relation:

C_v = Q / ndt​

Where,
Q ⇒ heat
'n' ⇒ no. of moles
C_v ⇒ Molar Heat capacity at constant volume
dT ⇒ Change in temperature

Now, for an isochoric process;

Q = dE_int​

∴ dE_int = nC_vdT → (2)​

Substituting dE_int from eq.(2) into eq.(1)

⇒ W = nC_vdT​

As thermodynamic work is given as W = -PdV

∴ -PdV = nC_vdT
⇒ PdV = - nC_vdT → (3)​

Now, writing equation of state of the gas in differential form as;

d(PV) = d(nRT)​

⇒ PdV + VdP = nRdT​

Using eq.(3)

⇒ - nC_vdT + VdP = nRdT
⇒ VdP = nC_vdT + nRdT​

Using the relation C_p = C_v + R into the above equation

⇒ VdP = nC_pdT → (4)​

Now, dividing eq.(4) by eq.(3)

⇒ VdP / PdV = -C_p / C_v
or VdP / PdV = -γ​

where, γ ⇒ ratio of molar heat capacities

Rearranging the above equation:

dP / P = -γ dV / V​

Integrating both sides with the initial state 'i' and final state 'f' being the lower and upper limits respectively, gives;

∫ dP / P = -γ ∫ dV / V​

⇒ P_iV_i^γ = P_fV_f^γ​

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 dE_int 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)

 

[drvrm]

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)

For an isochoric process the work done is always zero.
A typical example of an isochoric process is addition or removal of heat from a closed system.
The volume remains constant but temperature and pressure change according to the process.
thus leading to change in internal energy.

An adiabatic system is perfectly insulated from external environment and there is no heat transfer in or out of the system.
Work done in an adiabatic process completely results in change in the internal energy of the system.
so the two processes can be related.The heat capacity ratio(gamma) for an ideal gas can be related to the degrees of freedom of a molecule of gas

gamma = 1 + 2/(degree of freedom )
for a monoatomic gas, degree of freedom is 3 so its value is (1 +2/3) = 1.67
and for diatomic gas it (gamma) goes to 1.4 as the gas has five degrees of freedom.
This ratio becomes important when one uses the gas in a cyclic process for conversion of heat into work. and gas is taken as working substance like in a heat ( carnot engine.) engine.