Answer
346.5k+ views
Hint:In order to solve this question we need to understand surface tension and heat. Consider a water bubble inside water, it experiences force from all directions but if we consider water bubbles on surface of water then it experience force from only three directions, not from above as above the bubble there is no water, so a unbalanced downward force acts on bubble and due to this tension developed on the surface, this is known as surface tension. It is mathematically defined as the ratio of force per unit length. Heat is defined as energy which flows due to temperature difference.
Complete step by step answer:
Consider a water bubble of spherical shape having radius $r$.Given the surface tension is, $T$ and according to the problem, $T = \sqrt {\pi \rho } $.
So a force due to surface tension is given by, $F = Tl$
Here $l$ is length of bubble which is equal to, $l = 2\pi r$
So force is, $F = 2\pi rT$
Since the water level rises by an amount of height, “h”.So Volume of water, as it forms a cylinder of height “h” and radius “r” so, $V = \pi {r^2}h$.Since density of water is, $\rho $.So mass of water which rise is, $m = \rho V$
$m = \pi {r^2}h\rho $
So force of gravity is,
${F_g} = mg$
$\Rightarrow {F_g} = \pi {r^2}h\rho g$
Considering water bubble to at equilibrium we equate both forces, we get, $F = {F_g}$
$2\pi rT = \pi {r^2}h\rho g$
Putting values we get, $2\pi r\sqrt {\pi \rho } = \pi {r^2}h\rho g$
$h = \dfrac{{2\pi r\sqrt {\pi \rho } }}{{\pi {r^2}\rho g}}$
$\Rightarrow h = \dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} $
So work done by force due to surface tension in raising height “h” is, ${W_1} = Fh$
Putting values we get, ${W_1} = 2\pi rTh$
${W_1} = 2\pi r(\sqrt {\pi \rho } )(\dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} )$
$\Rightarrow {W_1} = \dfrac{{4{\pi ^2}}}{g}$
Also Potential energy stored by gravity is , ${U_g} = \dfrac{{mgh}}{2}$
${U_g} = \dfrac{{(\pi {r^2}h)\rho gh}}{2}$
$\Rightarrow {U_g} = \pi {r^2}\rho g{h^2}(\dfrac{1}{2})$
$\Rightarrow {U_g} = \pi {r^2}\rho g{(\dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} )^2}(\dfrac{1}{2})$
$\Rightarrow {U_g} = \dfrac{{2{\pi ^2}\rho g{r^2}}}{{{r^2}{g^2}\rho }}$
$\Rightarrow {U_g} = \dfrac{{2{\pi ^2}}}{g}$
So heat evolved is, given as, $H = {W_1} - {U_g}$
Putting values we get, $H = \dfrac{{4{\pi ^2}}}{g} - \dfrac{{2{\pi ^2}}}{g}$
$\therefore H = \dfrac{{2{\pi ^2}}}{g}$
So the amount of heat evolved in joules is, $H = \dfrac{{2{\pi ^2}}}{g}\,J$.
Note: It should be remembered that gravity will act only on the center of mass of water raised which is at height $\dfrac{h}{2}$ so , potential energy stored in water contains a half factor. Also energy is released in form of heat if the eater raises, but if water falls in the capillary tube then heat would be required to do work against gravity.
Complete step by step answer:
Consider a water bubble of spherical shape having radius $r$.Given the surface tension is, $T$ and according to the problem, $T = \sqrt {\pi \rho } $.
So a force due to surface tension is given by, $F = Tl$
Here $l$ is length of bubble which is equal to, $l = 2\pi r$
So force is, $F = 2\pi rT$
Since the water level rises by an amount of height, “h”.So Volume of water, as it forms a cylinder of height “h” and radius “r” so, $V = \pi {r^2}h$.Since density of water is, $\rho $.So mass of water which rise is, $m = \rho V$
$m = \pi {r^2}h\rho $
So force of gravity is,
${F_g} = mg$
$\Rightarrow {F_g} = \pi {r^2}h\rho g$
Considering water bubble to at equilibrium we equate both forces, we get, $F = {F_g}$
$2\pi rT = \pi {r^2}h\rho g$
Putting values we get, $2\pi r\sqrt {\pi \rho } = \pi {r^2}h\rho g$
$h = \dfrac{{2\pi r\sqrt {\pi \rho } }}{{\pi {r^2}\rho g}}$
$\Rightarrow h = \dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} $
So work done by force due to surface tension in raising height “h” is, ${W_1} = Fh$
Putting values we get, ${W_1} = 2\pi rTh$
${W_1} = 2\pi r(\sqrt {\pi \rho } )(\dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} )$
$\Rightarrow {W_1} = \dfrac{{4{\pi ^2}}}{g}$
Also Potential energy stored by gravity is , ${U_g} = \dfrac{{mgh}}{2}$
${U_g} = \dfrac{{(\pi {r^2}h)\rho gh}}{2}$
$\Rightarrow {U_g} = \pi {r^2}\rho g{h^2}(\dfrac{1}{2})$
$\Rightarrow {U_g} = \pi {r^2}\rho g{(\dfrac{2}{{rg}}\sqrt {\dfrac{\pi }{\rho }} )^2}(\dfrac{1}{2})$
$\Rightarrow {U_g} = \dfrac{{2{\pi ^2}\rho g{r^2}}}{{{r^2}{g^2}\rho }}$
$\Rightarrow {U_g} = \dfrac{{2{\pi ^2}}}{g}$
So heat evolved is, given as, $H = {W_1} - {U_g}$
Putting values we get, $H = \dfrac{{4{\pi ^2}}}{g} - \dfrac{{2{\pi ^2}}}{g}$
$\therefore H = \dfrac{{2{\pi ^2}}}{g}$
So the amount of heat evolved in joules is, $H = \dfrac{{2{\pi ^2}}}{g}\,J$.
Note: It should be remembered that gravity will act only on the center of mass of water raised which is at height $\dfrac{h}{2}$ so , potential energy stored in water contains a half factor. Also energy is released in form of heat if the eater raises, but if water falls in the capillary tube then heat would be required to do work against gravity.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)