= Solution
For a spherical <micelle> of <sodium dodecyl sulfate> with $n$ molecules and a sharp tail-filled core of <radius> $R$, volume and head-group area give $nv=4\pi R^3/3$ and $na_0=4\pi R^2$. Dividing gives the <spherical micelle geometry>
$$
\boxed{R=\frac{3v}{a_0}=1.84\,\mathrm{nm},\qquad n=\frac{4\pi R^2}{a_0}\simeq75.}
$$
There is a consistency limit: $R$ exceeds the supplied maximum tail length $1.67\,\mathrm{nm}$. Equivalently the <amphiphile packing parameter> is $v/(a_0l_c)=0.368>1/3$, outside the strict spherical-core bound. Thus the requested spherical assumption gives these estimates, but the three molecular dimensions cannot all be exact within that ideal sharp-core geometry. Effective head area, core structure or aggregate shape must adjust; silently replacing $R$ by $l_c$ would no longer satisfy both packing equations.
Back to article page
