Consider a bar where the regular clientele are avid chain smokers. A group of these regulars come to the bar and start chain smoking at $10$:$00$ pm; no one was smoking in the bar prior to their arrival. They emit $900$ mg per hour of particulate matter from their cigarettes. The bar has a ventilation system with an airflow rate of $45$ m$3$ per hour. The volume of the bar is $100$ m$3.($a$)$ At $11$:$00$ pm$,$ what is the concentration of particulate matter in the bar, in mg per m$3?($b$)$ What is the characteristic time for the smoke concentration to reach a steady state level? $($c$)$ Non$-$smokers in the bar might be persuaded to support a smoking ban if particulate matter concentrations exceed $15$ mg per m$3.$ What is the minimum particulate matter emission rate that would drive them to this decision by the time they leave the bar at $1$:$00$ am$?$