Final answer:
The mass of the unknown object is 20632.65 kg. Calculate the work done by the gas as it expands and equate it to the gravitational potential energy. Given the work done is 202,650 J and using the formula for gravitational potential energy, the unknown mass is approximately 20,658 kg.
Step-by-step explanation:
To find the mass of the unknown object, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature. Rearranging the equation, we have n = PV / RT. Given that the final state of the gas has a pressure of 2.0 atm, a volume of 2.0 m³, and a temperature of 600K, and the initial state has a pressure of 2.0 atm, a volume of 1.0 m³, and a temperature of 300K, we can calculate the number of moles of gas in the system:
n = (2.0 atm * 1.0 m³) / (8.314 J/(mol·K) * 300 K) = 0.026 mol
Since the unknown object lifts 1.0 meter and does work against the force of gravity, the work done is equal to the change in potential energy of the unknown object. We can calculate the work done using the formula: work = force * distance. The force is equal to the weight of the object, which is given by the formula: force = mass * gravitational acceleration. Rearranging the formulas, we have mass = work / (gravitational acceleration * distance). Plugging in the values, we have:
mass = (force * distance) / (gravitational acceleration * distance) = force / gravitational acceleration = (P * A) / g
where A is the cross-sectional area of the piston and g is the acceleration due to gravity. Since the pressure on both sides of the piston is the same, the force is equal to the pressure times the area of the piston: force = pressure * area. Plugging in the values, we have:
mass = (P * A) / g = (2.0 atm * 101325 Pa/atm * 1 m²) / (9.8 m/s²) = 20632.65 kg