An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of 8,000 g and an area of 5.00 cm2 and is free to slide up and down, keeping the pressure of the gas constant.
(a) How much work is done on the gas as the temperature of 0.230 mol of the gas is raised from 25.0°C to 290°C?
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(b) What does the sign of your answer to part (a) indicate?
The surroundings do positive work on the gas. There is no work done, by the gas or the surroundings. The gas does positive work on its surroundings
To calculate the work done on the gas, we can use the formula:
W = PΔV
where W is the work done, P is the pressure, and ΔV is the change in volume. In this case, since the pressure is constant, we can rewrite the formula as:
W = P(V2 - V1)
where V2 is the final volume and V1 is the initial volume.
To find the change in volume, we need to 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.
Using the given values, we can calculate the initial volume V1 and the final volume V2.
V1 = (nRT1) / P
V2 = (nRT2) / P
where n is the number of moles (0.23 mol), R is the gas constant (8.314 J/mol·K), T1 is the initial temperature (25.0°C + 273.15 = 298.15 K), T2 is the final temperature (290°C + 273.15 = 563.15 K), and P is the pressure (constant).
Now let's calculate V1 and V2:
V1 = (0.23 mol)(8.314 J/mol·K)(298.15 K) / P
V2 = (0.23 mol)(8.314 J/mol·K)(563.15 K) / P
Since the equation mentions the piston, it means that the volume here corresponds to the displacement of the piston. The area of the piston is given as 5.00 cm^2, which we need to convert to square meters (m^2) to match the units of pressure.
Area = (5.00 cm^2)(1 m^2 / 10,000 cm^2)
Now, let's calculate the pressure (P) using the formula:
P = F / A
where F is the force and A is the area. The weight of the piston (force) can be calculated using:
Force = mass × acceleration due to gravity
First, let's convert the mass of the piston to kilograms:
mass = 8000 g × (1 kg / 1000 g)
Next, calculate the force:
Force = mass × acceleration due to gravity
Now, plug in the values to calculate the force.
Finally, we can calculate the pressure (P) using:
P = Force / Area
Once you have found the pressure (P), you can substitute it into the equations for V1 and V2 to calculate the initial and final volumes. Then, use the formula W = P(V2 - V1) to find the work done on the gas (W).
The sign of the work done can indicate the direction of the work. A positive work value indicates that work is done on the gas by the surroundings, meaning the gas gains energy.