Question

Applying First Law of Thermodynamics obtain an adiabatic relation between pressure and volume​

Answer 1

Answer:

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; .

dEint = W → (1)where, ΔEint ⇒ change in internal energy and W ⇒ workdone

Then, they used the relation: Cv = Q / ndtWhere, 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 = nCvdTAs thermodynamic work is given as W = -PdV ∴ -PdV = nCvdT ⇒ PdV = - nCvdT → (3)

Now, writing equation of state of the gas in differential form as; d(PV) = d(nRT)⇒ PdV + VdP = nRdTUsing eq.(3) ⇒ - nCvdT + VdP = nRdT ⇒ VdP = nCvdT + nRdTUsing 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 = -γwhere, γ ⇒ ratio of molar heat capacities

Rearranging the above equation: dP / P = -γ dV / VIntegrating both sides with the initial state 'i' and final state 'f' being the lower and upper limits respectively, gives; ∫ dP / P = -γ ∫ dV / V⇒ PiViγ = PfVfγwhich 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)

Answer 2

Answer:

Heya mate here your answer is:-⬇️

A/C to 1st law of thermodynamics;

dQ=dU+PdV

For adiabatic process,dQ=0

So dU+PdV=0

or, dU=-PdV

When the expansion takes place,volumeof the gas increases i.e.dV=+ve,hemce dU=+ve;

i.e.internal energy of the gas decreases and hence the tempreture falls.

If the gas is compressed,volume of the gas decreases,i.e.dV=-ve,hencedU=+ve,i.e.

internal energy of the gas increase and hence its tempreture increases.

⬆️hope it helps you⬆️

✌thank you✌