A movable piston having a mass of 7.26 kg and a cross-sectional area of 5.26 cm2 traps 0.250 moles of an ideal gas in a vertical cylinder. If the piston slides without friction in the cylinder, how much work will the gas do when its temperature is increased from 20.5oC to 286oC?
To find the work done by the gas when its temperature is increased, we need to use the formula for work done by a gas:
Work = Pressure x Change in Volume
To calculate the change in volume, 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, R is the ideal gas constant, and T is the temperature in Kelvin.
First, we need to convert the temperatures from Celsius to Kelvin:
Initial temperature (T1) = 20.5 + 273.15 = 293.65 K
Final temperature (T2) = 286 + 273.15 = 559.15 K
Next, we need to find the initial and final volumes.
The initial volume (V1) is not given directly, but we can find it using the ideal gas law:
V1 = (n1RT1) / P
where R is the ideal gas constant and P is the initial pressure.
The final volume (V2) can be calculated using the same equation:
V2 = (n2RT2) / P
where n2 is the number of moles of gas at the final temperature, and P is the final pressure.
Since the piston slides without friction, we can assume that the pressure remains constant. Thus, P1 = P2 = P.
Now, let's calculate the initial and final volumes:
V1 = (0.250 mol x R x 293.65 K) / P
V2 = (0.250 mol x R x 559.15 K) / P
The cross-sectional area of the piston is given as 5.26 cm^2. Thus, we can calculate the initial and final volumes using the formula:
V1 = initial area x initial height = 5.26 cm^2 x h1
V2 = initial area x final height = 5.26 cm^2 x h2
Since the mass of the piston is given, we can calculate the heights by dividing the mass by the product of the cross-sectional area and the acceleration due to gravity:
h1 = (7.26 kg) / (5.26 cm^2 x 9.8 m/s^2)
h2 = h1 (As the piston slides without friction)
Now we have the initial and final volumes in terms of the heights:
V1 = (5.26 cm^2 x h1)
V2 = (5.26 cm^2 x h2)
Finally, substituting the values in the work equation:
Work = P x (V2 - V1)
Work = P x [(5.26 cm^2 x h2) - (5.26 cm^2 x h1)]