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I have a little query concerning the derivation of PV

^{γ}= constant. In my text book of Physics, first they give the equation for adiabatic process using the first law of Thermodynamics, as;

**d**E

_{int}= W → (1)

where,

ΔE

_{int}⇒ change in internal energy and W ⇒ workdone

Then, they used the relation:

C

_{v}= Q / n**d**tQ ⇒ heat

'n' ⇒ no. of moles

C

_{v}⇒ Molar Heat capacity at constant volume

**d**T ⇒ Change in temperature

Now, for an isochoric process;

Q =

∴**d**E_{int}**d**E

_{int}= nC

_{v}

**d**T → (2)

Substituting dE

_{int}from eq.(2) into eq.(1)

⇒ W = nC

_{v}**d**TAs thermodynamic work is given as W = -P

**d**V

∴ -P

⇒ P

**d**V = nC_{v}**d**T⇒ P

**d**V = - nC_{v}**d**T → (3)Now, writing equation of state of the gas in differential form as;

**d**(PV) =

**d**(nRT)

**d**V + V

**d**P = nR

**d**T

⇒ - nC

⇒ V

_{v}**d**T + V**d**P = nR**d**T⇒ V

**d**P = nC_{v}**d**T + nR**d**TUsing the relation C

_{p}= C

_{v}+ R into the above equation

⇒ V

**d**P = nC_{p}**d**T → (4)Now, dividing eq.(4) by eq.(3)

⇒ V

or V

**d**P / P**d**V = -C_{p}/ C_{v}or V

**d**P / P**d**V = -**γ****γ**⇒ ratio of molar heat capacities

Rearranging the above equation:

**d**P / P = -

**γ d**V / V

∫

**d**P / P = -**γ**∫**d**V / V⇒ P

_{i}V_{i}^{γ}= P_{f}V_{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)