A spherical unicellular organism relies entirely on simple diffusion across its cell surface membrane to supply its cytoplasm with oxygen. The rate of oxygen supply into the cell (RinR_{\text{in}}Rin, in mol s−1\text{mol}\,\text{s}^{-1}mols−1) is proportional to its surface area (AAA), whereas its metabolic oxygen demand (DmetD_{\text{met}}Dmet, in mol s−1\text{mol}\,\text{s}^{-1}mols−1) is proportional to its volume (VVV).
These relationships are defined by:
Rin=k⋅A R_{\text{in}} = k \cdot A Rin=k⋅A Dmet=m⋅V D_{\text{met}} = m \cdot V Dmet=m⋅Vwhere:
- A=4πr2A = 4\pi r^2A=4πr2 and V=43πr3\displaystyle V = \frac{4}{3}\pi r^3V=34πr3 (where r r\,r is the radius of the cell in μm\mu\text{m}μm)
- k=1.2×10−6 mol μm−2 s−1k = 1.2 \times 10^{-6}\,\text{mol}\,\mu\text{m}^{-2}\,\text{s}^{-1}k=1.2×10−6molμm−2s−1
- m=9.0×10−8 mol μm−3 s−1m = 9.0 \times 10^{-8}\,\text{mol}\,\mu\text{m}^{-3}\,\text{s}^{-1}m=9.0×10−8molμm−3s−1
What is the maximum radius (rmaxr_{\text{max}}rmax) this cell can reach before its metabolic demand exceeds its rate of oxygen supply?
13.3 μm13.3\,\mu\text{m}13.3μm
40.0 μm40.0\,\mu\text{m}40.0μm
4.4 μm4.4\,\mu\text{m}4.4μm
120.0 μm120.0\,\mu\text{m}120.0μm