Answer :
Final answer:
The mass of water that rises in a capillary tube of radius '2r' being (3) 2M. is the same as the mass M that rises in a capillary tube of radius 'r'. This is because, while the height of the liquid column will be halved when the radius is doubled, the volume — and consequently the mass — remains unchanged.
Explanation:
The mass of water that rises in a capillary tube due to capillary action is dependent on the capillary rise equation, which is given as 2T cos ϴ h = rpg, where h is the height of the liquid rise, T is the surface tension, ϴ is the contact angle, r is the tube radius, p is the density of the liquid, and g is the acceleration due to gravity. When the contact angle is 0°, which is typical for water in a glass tube, the height of water rise in the capillary is inversely proportional to the radius of the tube.
Given that mass M is the mass of water that rises in a capillary of radius r, and the volume of the liquid column is proportional to the square of the radius times the height (V = πr2h), we can deduce that when the radius is doubled, the height of the water column will be halved due to the inverse relationship. Thus, the volume of water in a tube with radius 2r will be (π(2r)2(h/2)) which simplifies to πr2h, the same volume as the original mass M. Therefore, the mass of water that will rise in a capillary tube with radius 2r is M.
