A specific volume of dry air is expanded to 6 times it volume at standard pressure and temperature under Adiabatic condition and isothermal condition in each cases evaluate the final temperature and pressure.Given that the specific gas ratio gamma=1.40 for air.
To solve this problem, we can use the equations for adiabatic and isothermal expansion.
For adiabatic expansion, we can use the following equations:
1) For volume expansion: (V2/V1) = (P1/P2)^(1/gamma)
2) For temperature change: (T2/T1) = (V1/V2)^(gamma-1)
For isothermal expansion, the equations are as follows:
1) For volume expansion: (V2/V1) = (P1/P2)
2) For temperature change: (T2/T1) = 1
Given that the initial volume (V1) is being expanded to 6 times its volume, we can set the final volume (V2) as 6V1.
1) Adiabatic Expansion:
Using equation 1):
6 = (P1/P2)^(1/1.40)
Taking the logarithm of both sides:
log(6) = (1/1.40)log(P1/P2)
log(P1/P2) = log(6)^(1.40)
log(P1/P2) ≈ 0.9816
Taking the antilogarithm:
P1/P2 ≈ 10^0.9816
P1/P2 ≈ 9.10
Using equation 2):
(T2/T1) = (V1/V2)^(gamma-1)
(T2/T1) = (1/6)^(0.40)
(T2/T1) ≈ 0.589
T2 = T1 * 0.589
2) Isothermal Expansion:
Using equation 1):
6 = (P1/P2)
P2 = P1/6
Using equation 2):
(T2/T1) = 1
T2 = T1
Therefore, the final temperature and pressure for each case are as follows:
Adiabatic Expansion:
Final Temperature (T2) ≈ T1 * 0.589
Final Pressure (P2) ≈ 9.10 * P1
Isothermal Expansion:
Final Temperature (T2) = T1
Final Pressure (P2) = P1/6