Two kilograms of an ideal gas undergoes an isentropic process from 670 KPag and a volume of 0.80 cubic meter to a final volume of 0.51 cubic meter. If constant pressure is 0.5215 and constant volume is 0.3152 KJ/kgK, what are (a) final temperature, (b)final absolute pressure, (c)change of enthalpy, (d) Wk.

To find the final temperature (T2), final absolute pressure (P2), change in enthalpy (ΔH), and the work done (Wk) in this isentropic process, we can use the ideal gas equation, the definition of specific heat capacity at constant volume, and the definition of specific heat capacity at constant pressure.

Given:
Initial temperature (T1) = 670 K
Initial absolute pressure (P1) = 670 kPa
Initial volume (V1) = 0.80 m^3
Final volume (V2) = 0.51 m^3
Constant volume specific heat capacity (Cv) = 0.3152 kJ/kgK
Constant pressure specific heat capacity (Cp) = 0.5215 kJ/kgK
Mass of the gas (m) = 2 kg

(a) To find the final temperature (T2):
Since the process is isentropic (adiabatic and reversible), we can use the relationship between temperature and volume for an adiabatic process:

(T1 / T2) = (V2 / V1)^(γ - 1)

where γ = Cp / Cv is the ratio of specific heat capacities.

Substituting the known values:
(670 / T2) = (0.51 / 0.80)^(0.5215 / 0.3152)
(670 / T2) = 0.6368^1.65693

Now, solve for T2:
T2 = 670 / (0.6368^1.65693)

(b) To find the final absolute pressure (P2):
For an isentropic process of an ideal gas, we can use the relationship between temperature and pressure:

(P1 / P2) = (T1 / T2)^(γ / (γ - 1))

Substituting the known values:
(P1 / P2) = (670 / T2)^(0.5215 / (0.5215 - 0.3152))

Now, solve for P2:
P2 = P1 / ((670 / T2)^(0.5215 / (0.5215 - 0.3152)))

(c) To find the change in enthalpy (ΔH):
ΔH = Cp * ΔT

Since the process is isentropic, ΔT = T2 - T1.

Substitute the known values:
ΔH = Cp * (T2 - T1)

(d) To find the work done (Wk):
Wk = ΔH - Q

Since the process is isentropic, Q = 0 (adiabatic process).

Therefore,
Wk = ΔH

Now, we can calculate the values of (a) final temperature (T2), (b) final absolute pressure (P2), (c) change in enthalpy (ΔH), and (d) work done (Wk) using the above equations and the given values.