Chapter 10, Problem 10.10P

10.10. A ventilation system has been designed for a large laboratory with a volume of 1100 m³. The volumetric flow rate of ventilation air is 700 m³/min at 22°C and 1 atm. (The latter two values may also be taken as the temperature and pressure of the room air.) A reactor in the laboratory is capable of emitting as much as 1.50 mol of sulfur dioxide into the room if a seal ruptures. An SO₂mole fraction in the room air greater than 1.0 × 10-6(1 ppm) constitutes a health hazard.

1. Suppose the reactor seal ruptures at a time t = 0, and the maximum amount of SO₂is emitted and spreads uniformly throughout the room almost instantaneously. Assuming that the air flow is sufficient to make the room air composition spatially uniform, write a differential SO₂balance, letting N be the total moles of gas in the room (assume constant) and x(t) the mole fraction of SO₂in the laboratory air. Convert the balance into an equation for dx/dt and provide an initial condition. (Assume that all of the SO₂emitted is in the room at t = 0.)
2. Predict the shape of a plot of x versus t. Explain your reasoning, using the equation of Part (a) in your explanation.
3. Separate variables and integrate the balance to obtain an expression for.x(t). Check your solution.
4. Convert the expression for x(t) into an expression for the concentration of SO₂in the room, Cso₂(mol SO₂/L). Calculate (i) the concentration of SO₂in the room two minutes after the rupture occurs, and (ii) the time required for the SO₂concentration to reach the “safe” level.
5. Why would it probably not yet be safe to enter the room after the time calculated in Part (d)? (Hint: One of the assumptions made in the problem is probably not a good one.)