An air bubble rises from the button of a lake to its upper surface.The diameters of the bubble at the bottom and the surface are 3.6mm and 4 mm respectively.Depth of the lake is 2.5m and the temperature at the upper surface is 40^@C. Find the temperature at the bottom of the lake. Ignore the change in density of water with height. (Atmospheric pressure =76 cmHg and g=980 cm.s^-2)

An air bubble rises from the button of a lake to its upper surface.The

1.	An air bubble rises from the button of a lake to its upper surface.The diameters of the bubble at the bottom and the surface are 3.6mm and 4 mm respectively.Depth of the lake is 2.5m and the temperature at the upper surface is 40^@C. Find the temperature at the bottom of the lake. Ignore the change in density of water with height. (Atmospheric pressure =76 cmHg and g=980 cm.s^-2)
Answer» Solution :PRESSURE at the BOTTOM of the lake `p_1=76 times 13.6 times 980+250 times 1times 980` `=(76 times 13.6+250) times 980 dyn.cm^-2` Volume of AIR BUBBLE at the bottom `V_1=4/3pi(0.18)^3 cm^3` Let the temperature at the bottom of the lake =`T_1` Now at the surface pressure `p_2=76 times 13.6 times 980 dyn.cm^-2`. Volume of air bubble `V_2=4/3pi (0.2)^3 cm^3`, temperature `T_2=273+40=313K` Now using `(p_1V_1)/T_1=(p_2V_2)/T_2`, we get `((76times13.6+250)times980times4/3pi(0.18)^3)/T_1=(76times13.6times980times4/3pi(0.2)^3)/313` or,`T_1=283.37K=283.37-273^@C=10.37^@C`