Problem $1(10pts)$ : The use of biofuel stoves for cooking has led to significant health problems in some lowresource countries. The stoves produce a large quantity of small particles $($PM$2\mathrm{.}5).$ Assume that an indoor space of $17\mathrm{.}5m^3$ has a stove producing particulate matter at a rate of $72\frac{g}{h}($for comparison, EPA certified wood stoves need to meet a particulate emissions limit of $4\mathrm{.}5\frac{g}{h}).$ Assume $1\%($by mass$)$ of the particulate matter emitted is PM$2\mathrm{.}5.$ Assume the decay rate constant for PM$2\mathrm{.}5$ in ambient air is $0\mathrm{.}42h^{-1}.$

a$)(5$pts$)$ For the situation described in this problem, derive $($from the conservation of mass equation$)$ a model for the steady$-$state concentration of $P{M}_{2\mathrm{.}5}$ in the indoor space $($i$.$e$.C=?).$ Define all your terms. For simplicity, you can assume that the ventilation air $($air coming in$)$ has no $P{M}_{2\mathrm{.}5}.$

b$)(5$pts$)$ One engineering solution to the sooty stove problem is to increase the ventilation of the room. What ventilation rate is required to reduce the $P{M}_{2\mathrm{.}5}$ concentration to the National Ambient Air Quality Standard $($NAAQS$)$ of $35\frac{g}{m^3}?$ Express your answer as $\frac{m^3}{h}$our and as air changes per hour $($ ACH $).$ Is this ACH value reasonable? You can reference the building ventilation recommendation from the Centers for Disease Control and Prevention $($CDC$)$ here $($look for "How Much Ventilation Is Enough?"$).$