Two vertical capillary tubes of radii r₁ and r₂ (where r₁ = 2r₂ ) are dipped into the same liquid of surface tension T and density . The liquid rises to equilibrium heights in both tubes. During this process, heat is released due to the difference between the work done by the surface tension and the increase in the gravitational potential energy of the liquid column. The ratio of the heat released during the capillar

Step-by-step solution

The heat released Q during capillary rise is the difference between the work done by the surface tension ( W ) and the increase in gravitational potential energy ( U ) of the liquid column. The work done by surface tension is W = (T 2 r ) h . The increase in potential energy is U = ( r^2 h ) g ( h 2 ) = r^2 h^2 g 2 . From the capillary rise formula, we have h = 2T g r , which gives g r h = 2T . Substituting this into the expression for U : U = 1 2 r h ( g r h) = 1 2 r h (2T ) = r T h = W 2 . Therefore, the heat rel