Find. The mass of granular activated carbon (GAC) required per day to achieve the target effluent concentration.
Approach. Compute the daily mass of xylenes that must be removed from a simple influent–effluent mass balance, then size the carbon from the Freundlich isotherm evaluated at the target effluent concentration $C_e$ — the standard equilibrium-adsorber sizing convention, since a well-run GAC bed is taken to breakthrough, at which point the carbon leaving service is essentially in equilibrium with the very effluent concentration the design targets.
Convert flow and find the mass of xylenes removed per day. $$Q = 10{,}000\ \text{gal/day}\times3.785\ \tfrac{\text{L}}{\text{gal}} = 37{,}854\ \text{L/day}$$ $$\dot m_{\text{removed}} = Q(C_0-C_e) = 37{,}854\ \text{L/day}\times(600-10)\ \text{mg/L} = \boxed{22{,}334{,}000\ \text{mg/day}} = 22.33\ \text{kg/day}$$
Carbon capacity at the target effluent concentration. Evaluating the Freundlich isotherm at $C_f = C_e = 10\ \text{mg/L}$: $$q_e = 51.3\,(10)^{0.187} = 51.3\times1.538 = \boxed{78.9\ \text{mg xylene/g carbon}}$$
Check: evaluating $q_e$ at the target EFFLUENT concentration (not the influent) is the standard, most-common convention for this class of exam problem — it represents the carbon at the point of breakthrough. Evaluating instead at $C_0=600$ mg/L (a fresh-bed, far-from-breakthrough state) would give a larger apparent capacity, $q=51.3(600)^{0.187}\approx137.9$ mg/g, and a proportionally smaller (non-conservative) carbon estimate of about 162 kg/day; the effluent-based figure above is the defensible design value.
