A system contains 0.15 m3 of a gas at a pressure of 3.8 bar and 150 C. It is expanded adiabatically till the pressure falls to 1 bar. The gas is then heated at a constant pressure till its change in enthalpy is 70 kJ. Determine the total work done.
Write a critical note on the conservation of energy principal for closed and open system
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To determine the total work done, we need to consider two steps in the process: the adiabatic expansion and the constant pressure heating.
Step 1: Adiabatic Expansion
In an adiabatic process, no heat is exchanged with the surroundings, so the change in internal energy of the system is equal to the work done.
The work done during adiabatic expansion can be calculated using the equation:
\(W_1 = \frac{{P_2V_2 - P_1V_1}}{{\gamma - 1}}\)
where:
\(W_1\) = work done during adiabatic expansion
\(P_1\) = initial pressure (3.8 bar)
\(V_1\) = initial volume (0.15 m³)
\(P_2\) = final pressure (1 bar)
\(V_2\) = final volume (unknown)
\(\gamma\) = heat capacity ratio (ratio of specific heats) which depends on the gas
To solve for \(V_2\), we need to know the heat capacity ratio (\(\gamma\)). The value of \(\gamma\) depends on the specific gas being used. Could you provide the specific gas?
To determine the total work done, we need to calculate the work done during the adiabatic expansion and the work done during the constant pressure heating.
1. Adiabatic Expansion:
During the adiabatic expansion, we can use the formula for adiabatic work done:
W = Cp * (T2 - T1) / (1 - γ)
Where:
W is the work done
Cp is the heat capacity at constant pressure
T1 is the initial temperature
T2 is the final temperature
γ is the heat capacity ratio (also known as the adiabatic index)
Since the gas is expanding adiabatically, no heat is transferred, so Cp = Cv (heat capacity at constant volume). The heat capacity ratio γ for a diatomic gas can be approximated as 1.4.
We can find the final temperature T2 using the adiabatic process equation:
P1 * V1^(γ) = P2 * V2^(γ)
Where:
P1 is the initial pressure
V1 is the initial volume
P2 is the final pressure
V2 is the final volume
We know the initial pressure P1 = 3.8 bar, and the initial volume V1 = 0.15 m^3.
We are given the final pressure P2 = 1 bar.
We can rearrange the adiabatic process equation to solve for V2:
V2 = (P1 / P2)^(1/γ) * V1
Substituting the values, we can find V2.
Once we have T2 and V2, we can calculate the work done during the adiabatic expansion using the adiabatic work formula.
2. Constant Pressure Heating:
During the constant pressure heating, the work done is given by the change in enthalpy:
W = ΔH
We are given that the change in enthalpy ΔH is 70 kJ.
So, to determine the total work done, we need to add the work done during the adiabatic expansion and the work done during the constant pressure heating:
Total work done = Work done during adiabatic expansion + Work done during constant pressure heating.