An ideal diatomic gas in a cylinder with a movable piston un

An ideal diatomic gas in a cylinder with a movable piston undergoes the cyclic process. Assume that the temperature is such that rotational modes are active(but vibration modes are not) and work is quasi-static compression expansion type.

1) For each of the four branches, calculate the work done on the gas, the heat added to the gas, and the change in the internal energy of the gas.

Adiabat P1 V2

Solution

For an ideal diatomic gas = 7/5 =1.4

PV = const

work done in an adiabatic process

W = òPdv = P₃V₂^1.4 *(V₂^-0.4 - V₁^-0.4)/-0.4

      as it is an adiabatic process Q =0 , no heat added.

U + W = 0

change in internal energy = -W

during the cycle from P2 to P3 Volume remain constant

work done W = 0

volume is constant

heat transferred = Cp*n*(T2-T1)

P2V2=nRT2

P1V2= nRT1

T2-T1 = V2(P2-P1)/nR

Q = C_pV2(P2-P1)/R

as W=0

change in internal energy U = -Q = C_pV2(P1-P2)/R

From V2 to V1 P is constant P3

Work done W = P3(V2-V1) on the gas.

volume is constant

heat transferred Q = C_vn*T

PV = nRT

P1(V2-V1) = nRT

Q = C_v P1(V2-V1)/R

change in internal energy

Q = U + W

U = C_v P1(V2-V1)/R - P3(V2-V1)

from P3 to P1 Volume is constant and

W=0

volume is constant

heat transferred = Cp*n*T

PV=nRT

T = V3(P3-P1)/nR

Q = C_pV1(P3-P1)/R

as W=0

change in internal energy U = Q = C_pV1(P3-P1)/R