Continuous reactor—yield from steady-state concentrations: at steady state the reactor contains X = 0.04 g·L^-1 biomass and S_out = 0.02 g·L^-1 phenol; the feed contains S_in = 0.10 g·L^-1 phenol. Assuming biomass in feed is negligible, what is the biomass yield Y_x/s (g biomass produced per g substrate consumed)?

Introduction:
In continuous cultures at steady state, yield coefficients are often estimated from the difference between inlet and outlet substrate concentrations and the steady biomass concentration, assuming negligible biomass in the feed. This provides a quick screen of culture efficiency without detailed kinetic modeling.

Given Data / Assumptions:

• Biomass concentration in reactor (and effluent) X = 0.04 g·L^-1.
• Substrate in feed S_in = 0.10 g·L^-1; in effluent S_out = 0.02 g·L^-1.
• Negligible product/byproduct carbon sinks; yield approximated from material differences.

Concept / Approach:
Substrate consumed per liter = S_in − S_out = 0.08 g·L^-1. Biomass present per liter is the result of that consumption at steady state. The apparent yield is Y_x/s ≈ X / (S_in − S_out) under the stated simplifications.

Step-by-Step Solution:
Step 1: Compute substrate consumed: 0.10 − 0.02 = 0.08 g·L^-1.Step 2: Take the ratio with biomass: Y_x/s = 0.04 / 0.08 = 0.5 g.g^-1.Step 3: Select 0.5 g.g^-1 from the options.

Verification / Alternative check:
This back-of-the-envelope method aligns with steady-state CSTR balances when maintenance and product formation are modest relative to growth-associated biomass formation.

Why Other Options Are Wrong:

• 0.4 or 0.2 g.g^-1: Miscompute the ratio.
• 1.0 or 5.0 g.g^-1: Implausibly high given the data and would require different S_out or X.

Common Pitfalls:
Ignoring non-growth-associated product formation or maintenance terms; for precise yields, full carbon balances are preferred.

Final Answer:
0.5 g.g^-1