A spherical cell relies entirely on simple diffusion across its outer membrane to supply oxygen for aerobic respiration. The maximum rate of oxygen diffusion into the cell is given by the equation:
J=P⋅A⋅ΔC J = P \cdot A \cdot \Delta C J=P⋅A⋅ΔCwhere P=2.0×10−5 m s−1P = 2.0 \times 10^{-5}\text{ m s}^{-1}P=2.0×10−5 m s−1 is the permeability coefficient of the membrane, A A\,A is the surface area of the cell, and ΔC=1.5×10−2 mol m−3\Delta C = 1.5 \times 10^{-2}\text{ mol m}^{-3}ΔC=1.5×10−2 mol m−3 is the concentration gradient across the membrane.
The rate of oxygen consumption by the cell is constant per unit volume: Q=1.2×10−1 mol m−3 s−1Q = 1.2 \times 10^{-1}\text{ mol m}^{-3}\text{ s}^{-1}Q=1.2×10−1 mol m−3 s−1.
What is the maximum radius RRR (in micrometres, μm\mu\text{m}μm) that this cell can reach before its oxygen consumption rate exceeds its maximum oxygen diffusion rate?
2.5 μm2.5\ \mu\text{m}2.5 μm
7.5 μm7.5\ \mu\text{m}7.5 μm
15.0 μm15.0\ \mu\text{m}15.0 μm
22.5 μm22.5\ \mu\text{m}22.5 μm