I have seen derivations of capillary height using force equations:
$$2\pi RT \cos{\theta} = \rho \pi R^{2}hg$$
which gives,
$$h = \frac{2T\cos{\theta}}{\rho gR}$$
Now, if we go about this another way, the pressure at the free surface $P_{\text{atm}}$ should be equal to the pressure inside the capillary tube, at the same level.
Or,
$$P_{\text{atm}} - \frac{2T}{R} + ρgh = P_{\text{atm}}$$
This will give:
$$h = \frac{2T}{\rho gR},$$ which is wrong.
What is the mistake in this alternate derivation?