= Solution
Here the <osmotic pressure> is a two-dimensional surface <pressure>, with units of <energy> per area. In the dilute regime, independent whole chains provide the translational <entropy>, so
$$
\boxed{\Pi\sim\frac{k_BT}{\ell^2}\frac\phi N,\qquad\phi\ll\phi^*.}
$$
In the <semidilute polymer solution>, use a correlation blob containing $g$ monomers. Within it the chain still follows the planar good-solvent law $\xi\sim\ell g^{3/4}$. Its area fraction is $\phi\sim g\ell^2/\xi^2\sim g^{-1/2}$, so $g\sim\phi^{-2}$ and $\xi\sim\ell\phi^{-3/2}$. There is about one independent thermal contribution per blob area. The <semidilute osmotic pressure of a two-dimensional polymer solution> is thus
$$
\boxed{\Pi\sim\frac{k_BT}{\xi^2}\sim\frac{k_BT}{\ell^2}\phi^3,\qquad\phi^*\ll\phi\ll1.}
$$
It is independent of total chain length once each chain contains many blobs. The two estimates agree at $\phi^*\sim N^{-1/2}$, where each gives $(k_BT/\ell^2)N^{-3/2}$.
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