The path followed by a fixed amount of a perfect gas during a reversible process is described by the following equation:
P + aV = c
where a and c are constants. In the initial state, P1 = 60 bar and V1 = 0.002 m3. In the final state, P2 = 20 bar and V2 = 0.004 m3.
(a) Determine the values of the constants a and c and state the units
During the process path (Path 1) described by the above equation, according to the documentation, heat of the amount 5000 J is added to the gas.
(b) Draw the path on a PV diagram.
Write down the expression for the interchange of work between a closed system and the surroundings. Calculate the values of the work interchange (W) for this path. Write down the First Law for closed systems and determine the change in internal energy for this path (DU).
(c) Use the general relation between enthalpy and internal energy to determine the change in enthalpy (DH) for Path 1. You should now be able to fill in the first row of the table below.
Now consider another process path (Path 2) between the same initial and final states. This consists of two steps. First, the pressure is reduced from 60 bar to 20 bar with the volume remaining constant at 0.002 m3. Then, the volume is increased from 0.002 m3 to 0.004 m3 with the pressure constant at 20 bar.
(d) Internal energy (U) and enthalpy (H) are known as functions of state. Explain what a function of state is and how this definition should help to fill in the respective entries for Path 2 in the table below.
(e) Draw Path 2 on the diagram and calculate the heat (Q) and work (W) interchanges between the system and surroundings. You should now be able to complete the table.
(f) Assuming that the gas behaves ideally, what happens to the temperature (increasing, decreasing or remaining constant) along the diagonal Path 1?
As a hint, you should obtain an expression for the derivative (dT/dV) and determine the sign of this derivative at the start and end of the diagonal path.
(g) Combine the differential forms of the First Law and the perfect gas EOS to show that
dQ = (cv/R + 1)PdV + cv/R vdP
What do you think might be the next thing to do, given that it is often good practice to be sceptical about any data that you might be given?