Consider the isothermal expansion of 4.50 mol of an ideal gas at 450 K from an initial pressure of 12.0 bar to a final pressure of 2.75 bar. Describe the process that will result in the greatest amount of work being done by the system and calculate work Describe the process that will result in the greatest amount of work done by the system with a constant external pressure and calculate w. What is the least amount of work done with no restrictions on the external pressure?
To find the process that will result in the greatest amount of work being done, we need to consider different expansion processes. Work is given by the equation:
W = -Pext∆V
where W represents work, Pext is the external pressure, and ∆V is the change in volume.
1. Isothermal Expansion with Constant External Pressure:
In this case, the external pressure remains constant throughout the expansion process. The work done by the system can be calculated using the equation mentioned above.
W = -Pext∆V
To find ∆V, 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 ideal gas constant, and T is the temperature.
So, the change in volume (∆V) is given by:
∆V = Vf - Vi
Substituting the ideal gas law into the equation we get:
∆V = (nRTf/Pf) - (nRTi/Pi)
Now we can calculate the work using the given values:
n = 4.50 mol
Ti = 450 K
Tf = 450 K
Pi = 12.0 bar
Pf = 2.75 bar
R = 0.0831 L bar/mol K
Substitute these values into the equation and solve for ∆V:
∆V = (4.50 mol * 0.0831 L bar/mol K * 450 K / 12.0 bar) - (4.50 mol * 0.0831 L bar/mol K * 450 K / 2.75 bar)
The obtained ∆V value can then be used to calculate the work (W) using the formula:
W = -Pext∆V
Here, external pressure (Pext) is equal to the final pressure (Pf) since it remains constant during the isothermal expansion. Substitute the values into the equation and calculate the work.
2. Isothermal Expansion without External Pressure Restriction:
In this case, there are no restrictions on the external pressure. Therefore, we can assume that the pressure inside and outside the system equalizes throughout the expansion process. It means that the external pressure (Pext) is equal to the final pressure (Pf) and also equal to the initial pressure (Pi) in this case.
Using the same equation for work:
W = -Pext∆V
Since Pext = Pf = Pi, we can simplify the equation to:
W = -Pi∆V
Substitute the values of Pi, n, R, Ti, Tf into the equation and calculate ∆V. Then use this value to calculate the work done by the system.
By following these steps, you can determine the process that results in the greatest amount of work being done by the system and calculate the work done in each case.