the radius of gyration

A random coil is a fuzzy blob with no sharp edge, so asking how big it is needs care — you cannot just read off a radius the way you would for a marble. The radius of gyration, written Rg, is the honest answer: it is the typical distance of the chain's own atoms from the coil's centre of mass, a single number that captures the overall spread of the tangle. Picture the coil as a little cloud of dots (the monomers); Rg tells you how far, on average, the dots sit from the middle of the cloud.

Precisely, Rg squared is the average of the squared distances of every segment from the centre of mass: Rg^2 = (1/N) times the sum over all N segments of |r_i minus r_cm|^2. For an ideal random coil this ties neatly to the end-to-end distance R by Rg^2 = R^2 / 6, so Rg = R divided by the square root of 6. Carry the earlier example forward: a chain with rms end-to-end distance 25 nm has Rg = 25 / sqrt(6) = about 10.2 nm. The definition is completely general — it works for any shape — and for a uniform solid sphere of radius a it gives Rg = sqrt(3/5) times a, which is why Rg is smaller than the object's physical outer radius. Like the coil size, Rg scales with chain length N as a power law, Rg proportional to N to the power nu: nu = 0.5 for an ideal chain, about 0.588 for a swollen chain in a good solvent, and 1/3 for a chain collapsed into a dense globule.

Rg matters above all because it is the size you can actually measure. Scattering methods — small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS), and static light scattering — do not photograph the chain; instead the way intensity falls off at small angles (the Guinier regime) reads out Rg directly. So Rg is the bridge between the abstract random-walk theory and the numbers a real instrument returns, and comparing measured Rg against the ideal-coil prediction tells you at a glance whether a chain is swollen, ideal, or collapsed — that is, how good the solvent is or whether the chain has folded up.