The lower end of a capillary tube of radius r is "placed vertically in water. Then with the rise of water in the capillary, heat evolved is +(π r2 h2 d g/n J) calorie, where h is the height of capillary rise, J is the joules mechanical equivalent of heat and d is the density of water. Find n.

Question

The lower end of a capillary tube of radius r is "placed vertically in water. Then with the rise of water in the capillary, heat evolved is +(π r2 h2 d g/n J) calorie, where h is the height of capillary rise, J is the joules mechanical equivalent of heat and d is the density of water. Find n. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

When the tube is placed vertically in water, water rises through height

h

given by

h = \frac{2 T c o s θ}{r d g}

Upward force

= 2 π r \times T cos θ

Work done by this force in raising water column through height

h

is given by

Δ W = (2 π r T cos θ) h = (2 π r h cos θ) T

= (2 π r h cos θ) (\frac{r h d g}{2 c o s θ}) = π r^{2} h^{2} d g

However, the increase in potential energy

Δ E_{P}

of the raised water column

= m g \frac{h}{2}

where

m

is the mass of the raised column of water.

∵ m = π r^{2} h d

So,

Δ E_{P} = (π r^{2} h d) (\frac{h g}{2}) = \frac{π r ^{2} h ^{2} d g}{2}

Further,

Δ W - Δ E_{p} = \frac{π r ^{2} h ^{2} d g}{2}

The part

(Δ W - Δ E_{P})

is used in doing work against viscous forces and frictional forces between water and glass surface and appears as heat. So heat released

= \frac{Δ W - Δ E _{P}}{J} = \frac{π r ^{2} h ^{2} d g}{2 J}

Q. The lower end of a capillary tube of radius r is "placed vertically in water. Then with the rise of water in the capillary, heat evolved is +nJπr2h2dg​ calorie, where h is the height of capillary rise, J is the joules mechanical equivalent of heat and d is the density of water. Find n.