= Solution
The numerical contour length is $L=Nb_0=10^{10}\times10^{-8}\,\mathrm m=100\,\mathrm m$, and $k_BT=4.14\times10^{-21}\,\mathrm J$. Begin with a three-dimensional <freely jointed chain> in which each chemical <monomer> is also an independent statistical link of length $b_0$. This neglects <excluded volume>, hydrodynamic effects on equilibrium statistics, and stretching of the chemical backbone.
An applied <force> $F$ gives a one-link <canonical partition function>
$$
Z_1=2\pi\int_{-1}^{1}e^{\xi u}du=4\pi\frac{\sinh\xi}{\xi},\qquad \xi=\frac{Fb_0}{k_BT}.
$$
The mean extension is $z=k_BT\partial_F\log Z_1^N=Nb_0(\coth\xi-1/\xi)$. Expanding the <Langevin function> at small <force> gives $z=Nb_0^2F/(3k_BT)+O(F^3)$. The <three-dimensional entropic chain stiffness> is thus
$$
\boxed{k_s=\frac{3k_BT}{Nb_0^2}=1.242\times10^{-14}\,\mathrm{N\,m^{-1}}.}
$$
The unforced root-mean-square end-to-end distance is $\sqrt N b_0=10^{-3}\,\mathrm m$, not the contour length. This response is entropic straightening of a fluctuating coil. Axial stretching of a straight material rod instead has stiffness $EA/L$, and is a different phenomenon. Aqueous solvent does not by itself prove ideal-chain statistics: a good-solvent <self-avoiding walk> has a different coil size and susceptibility. The estimate above states explicitly the model under which the supplied link data suffice.
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