Fed-batch calculation: A reactor starts with 2.0 L medium at 0.10 g/L biomass. You feed 1.0 L/h of substrate medium (no biomass) for 10 h. After 10 h, X = 0.20 g/L in the reactor. How much biomass was produced during this period?

Introduction / Context:
Fed-batch mass balances require careful attention to changing volume. Biomass produced equals the increase in reactor biomass mass (final mass minus initial mass), assuming no biomass in the feed and no harvest/bleed. This problem checks comfort with basic accumulation logic.

Given Data / Assumptions:

• Initial volume V0 = 2.0 L; initial biomass concentration X0 = 0.10 g/L.
• Feed rate F = 1.0 L/h for 10 h, biomass-free; thus ΔV_feed = 10 L.
• Final volume Vf = 2.0 + 10 = 12.0 L (no outflow).
• Final biomass concentration Xf = 0.20 g/L.

Concept / Approach:
Total biomass mass in the vessel is M = X * V. Biomass produced = M_final − M_initial − M_in + M_out. With no biomass in the feed (M_in = 0) and no outflow (M_out = 0), production reduces to ΔM = XfVf − X0V0.

Step-by-Step Solution:

Verification / Alternative check:
Sanity check: concentration doubled while volume increased sixfold; mass scaling to 2.4 g is consistent with both effects.

Why Other Options Are Wrong:

• 1.5 g and 3.0 g: inconsistent with mass balance at the stated concentrations/volumes.
• 6.0 g: far exceeds what 0.20 g/L at 12 L can support.
• 0.2 g: that is the initial mass, not the produced mass.

Common Pitfalls:
Forgetting the volume change in fed-batch; using ΔX instead of Δ(XV).

Final Answer:
2.2 g.