Without knowing the initial volume, we cannot calculate the exact ratio of the volume increase for the isothermal expansion. However, if we assume the initial and final volumes are equal, the ratio would be 1.
To determine the ratio of the volume increase, we need to consider the isothermal expansion of 2 moles of an ideal gas at 300K and the work performed by the machine.
First, let's calculate the maximum possible work that the machine could perform during the isothermal expansion. In an isothermal process, the work done by an ideal gas can be calculated using the formula:
Work = nRT ln(Vf/Vi)
Where:
- n is the number of moles of the gas
- R is the ideal gas constant
- T is the temperature in Kelvin
- Vi is the initial volume
- Vf is the final volume
Given that the work performed by the machine is 2 kJ and it performed 45% of the maximum possible work, we can calculate the maximum work:
Max Work = 2 kJ / 0.45 = 4.44 kJ
Now, using the formula for work done by an ideal gas, we can rearrange it to solve for the final volume:
Vf = Vi * e^(Work / (nRT))
Plugging in the values:
Vi = initial volume (unknown)
Vf = final volume (unknown)
n = 2 moles
R = ideal gas constant
T = 300K
Work = 4.44 kJ
Since we don't have the initial volume (Vi), we cannot calculate the exact ratio of volume increase without that information. However, if we assume that the initial and final volumes are equal, we can calculate the ratio:
Vf = Vi * e^(4.44 kJ / (2 * R * 300K))
If we assume Vi = Vf, the ratio of volume increase would be 1.