An ideal gas is enclosed in a cylinder with a movable piston on top. The piston has mass of 8000g and an area of 5.00cm^2 and is free to slide up and down, keeping the pressure of the gas constant. How much work is done as the temperature of 0.200 mol of the gas is raised from 200 degree Celcius to 300 degree Celcius?

To determine the work done when the temperature of an ideal gas is raised while keeping the pressure constant, we can use the formula for work done in a thermodynamic process. The work done (W) by the gas during an isobaric (constant pressure) process can be calculated using the equation:

Understanding the Variables

In this scenario, we have:

• Number of moles (n): 0.200 mol
• Initial temperature (T1): 200 °C (which is 473 K when converted to Kelvin)
• Final temperature (T2): 300 °C (which is 573 K when converted to Kelvin)
• Pressure (P): This will be derived from the weight of the piston.

Calculating the Pressure

The pressure exerted by the piston can be calculated using the formula:

P = F / A

Where:

• F: The force due to the weight of the piston, which is the mass (m) times the acceleration due to gravity (g).
• A: The area of the piston.

First, we need to convert the mass of the piston from grams to kilograms:

m = 8000 g = 8 kg

Now, using g = 9.81 m/s², we find the force:

F = m * g = 8 kg * 9.81 m/s² = 78.48 N

The area of the piston is given as 5.00 cm², which we convert to m²:

A = 5.00 cm² = 5.00 x 10^-4 m²

Now we can calculate the pressure:

P = F / A = 78.48 N / (5.00 x 10^-4 m²) = 156960 N/m² = 156.96 kPa

Calculating the Work Done

For an isobaric process, the work done can be calculated using the formula:

W = P * ΔV

Where ΔV is the change in volume. To find ΔV, we can use the ideal gas law:

PV = nRT

Rearranging gives us:

V = nRT / P

We will calculate the initial and final volumes:

Initial Volume (V1)

Using T1 = 473 K:

V1 = (0.200 mol) * (8.314 J/(mol·K)) * (473 K) / (156960 N/m²)

V1 ≈ 0.00558 m³

Final Volume (V2)

Using T2 = 573 K:

V2 = (0.200 mol) * (8.314 J/(mol·K)) * (573 K) / (156960 N/m²)

V2 ≈ 0.00677 m³

Change in Volume (ΔV)

Now we can find ΔV:

ΔV = V2 - V1 = 0.00677 m³ - 0.00558 m³ = 0.00119 m³

Final Calculation of Work Done

Now substituting back into the work formula:

W = P * ΔV = 156960 N/m² * 0.00119 m³ ≈ 187.83 J

Thus, the work done as the temperature of the gas is raised from 200 °C to 300 °C while keeping the pressure constant is approximately 187.83 Joules.