# Effect of temperature on capillary action

As a science teacher, I always explain kids about how water rises in a capillary tube: Capillary action occurs when the adhesion to the walls is stronger than the cohesive forces between the liquid molecules. We know that surface tension is the cohesive forces between molecules. First I thought that surface tension will decrease with a temperature increase and hence water will rise more in a capillary tube (More adhesion force will become dominant than cohesive force). But the formula says different story.

$$h = \frac{2\gamma \cos \theta}{\rho g r}$$

Neglecting the change of density with temperature, the height of water column should decrease with the surface tension with a temperature increase. But this is a contradictory to my earlier saying that the cohesive force will become less effective so adhesion force will attract more liquid upward. I wonder what is wrong with my logic.