Given information:
- Volume of the cylinder: 5 liters
- Pressure in the cylinder: 15 atm
- Volume of the evacuated vessel: 500 liters
- Time taken for the gas to enter the vessel: 30 minutes

Assumptions:
- The gas is considered to be an ideal gas.
- The temperature remains constant during the process.

Calculating the initial number of moles:
We can use the ideal gas law to calculate the initial number of moles of the gas in the cylinder.

PV = nRT

Where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature (in Kelvin)

Given:
P = 15 atm
V = 5 liters
R = 0.0821 L·atm/(mol·K)

We know that the temperature remains constant, so we can assume the initial temperature is the same as the final temperature. Therefore, we can cancel out the temperature terms from the equation.

(15 atm) * (5 liters) = n * (0.0821 L·atm/(mol·K))

n = (15 atm * 5 liters) / (0.0821 L·atm/(mol·K))
n ≈ 91.5 mol

Calculating the final pressure:
When the gas enters the evacuated vessel, its volume increases from 5 liters to 500 liters. Since the number of moles remains constant, we can use the ideal gas law again to calculate the final pressure.

PV = nRT

Given:
V = 500 liters
n = 91.5 mol
R = 0.0821 L·atm/(mol·K)

We can assume the final temperature is the same as the initial temperature.

P * 500 liters = 91.5 mol * (0.0821 L·atm/(mol·K)) * T

We can rearrange the equation to solve for P:
P = (91.5 mol * 0.0821 L·atm/(mol·K) * T) / 500 liters

Since the temperature remains constant, we can substitute it with a constant value, such as 1 (for simplicity).

P = (91.5 mol * 0.0821 L·atm/(mol·K) * 1) / 500 liters
P ≈ 0.149 atm

Calculating the work done:
The work done during the expansion of an ideal gas can be calculated using the formula:

Work = -PΔV

Where:
Work = work done (in L·atm)
P = pressure (in atm)
ΔV = change in volume (in liters)

Since the gas expands from 5 liters to 500 liters, the change in volume is:

ΔV = 500 liters - 5 liters
ΔV = 495 liters

Substituting the values into the formula, we get:

Work = -(0.149 atm) * (495 liters)
Work ≈ -73.755 L·atm

Since the work done is negative, it means work is done on the system (gas) by the surroundings. Therefore, the

Bcz, it is free expansion so, the pressure is zero and no met work done is also "0"