a piston cylinder device initially contains 10lb of acetylene at 70F. the piston is free to move to maintain 1000 psia on the gas. how much WORK is done by the gas as it is headed to 300F. (draw pv diagram)
To determine the work done by the gas, we can use the formula:
Work (W) = ∫PdV
where P is the pressure and dV is the change in volume.
First, let's draw a P-V diagram to visualize the process. The x-axis represents the volume (V), and the y-axis represents the pressure (P). We know that the initial pressure is 1000 psia, and the final temperature is 300F. Now, we need to find the final pressure to complete the diagram.
To find the final pressure, we can use the ideal gas law:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. We can rearrange the equation as follows:
P1V1 / T1 = P2V2 / T2
We know P1 = 1000 psia, V1 initially is unknown, T1 = 70°F, P2 is unknown, V2 is unknown, and T2 = 300°F. To use the ideal gas equation, we need to convert the temperatures to Kelvin:
T1 (K) = (70 + 459.67) Rankine
T2 (K) = (300 + 459.67) Rankine
Once we have P2, we can proceed to calculate the work done by integrating the equation:
W = ∫PdV
To evaluate this integral, we need an expression for pressure as a function of volume. For this, we can use the ideal gas equation:
PV = nRT
Dividing by V, we obtain:
P = nRT / V
To determine n, the number of moles of acetylene, we need to know the molecular weight:
n = m / M
where m is the mass of acetylene and M is the molecular weight of acetylene (C2H2).
Given that the device initially contains 10 lb of acetylene, we need to verify the molecular weight from a reference source. Assume the molecular weight of acetylene is 26 lb/lbmol.
Substituting the values into the equations, we can calculate the work done by integrating the equation:
W = ∫PdV
Integrating the equation involves finding the equation of the curve that represents the process path on the P-V diagram. Once we have the equation of this curve, we can integrate it within the desired volume limits to determine the work done.