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A vertical capillary of length L is sealed at its top end. When its open lower end touches a liquid surface, the liquid rises inside to a height h. The liquid density is rho, the inner diameter of the capillary is d, the contact angle is theta, and the atmospheric pressure is p0. Find the surface tension of the liquid.
- sigma = (d/4)*(p0*h/(L - h) + rho*g*h)/cos(theta)
- sigma = (d/4)*(p0*h/(L - h) - rho*g*h)/cos(theta)
- sigma = (d/4)*(p0*(L - h)/h - rho*g*h)/cos(theta)
- sigma = (d/4)*(p0/(L) - rho*g*h)/cos(theta)
Correct answer: sigma = (d/4)*(p0*h/(L - h) + rho*g*h)/cos(theta)
Solution
Trapped air initially had length L at pressure p0. After rise h it occupies length (L - h), so by Boyle's law its pressure becomes p = p0*L/(L - h). Pressure balance for the raised column: capillary (Laplace) pressure 4*sigma*cos(theta)/d = (p - p0) + rho*g*h = p0*[L/(L-h) - 1] + rho*g*h = p0*h/(L-h) + rho*g*h. Solving: sigma = (d/4)*(p0*h/(L-h) + rho*g*h)/cos(theta).
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