A spherical air bubble is rising from the depth of a lake where the pressure is 'P' atm and the temperature is 'T' Kelvin. The percentage increase in its radius when it comes to the free surface of the lake will be. (Assume temperature and pressure at the surface are respectively P/4 and 2T Kelvin)

Explanation

Ideal Gas Law:The ideal gas law states: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.Since the number of moles of air in the bubble remains constant, we can write: (P1V1)/T1 = (P2V2)/T22. Given Values:* P1 = P atm (pressure at depth) * T1 = T K (temperature at depth) * P2 = P/4 atm (pressure at surface) * T2 = 2T K (temperature at surface) * V1 = volume of the bubble at depth * V2 = volume of the bubble at surface3. Substituting Values into Ideal Gas Law:** * (PV1)/T = (P/4 * V2) / (2T)4. Simplifying the Equation:** * PV1/T = PV2 / 8T * V1/1 = V2/8 * V2 = 8V15. Volume of a Sphere:** * V = (4/3)πr³, where r is the radius. * V1 = (4/3)πr1³ (volume at depth) * V2 = (4/3)πr2³ (volume at surface)6. Substituting Volume Expressions:** * (4/3)πr2³ = 8 * (4/3)πr1³ * r2³ = 8r1³7. Finding the Relationship between Radii:** * Taking the cube root of both sides: r2 = 2r18. Calculating the Increase in Radius:** * Increase in radius = r2 - r1 = 2r1 - r1 = r19. Calculating the Percentage Increase:** * Percentage increase = (increase in radius / original radius) * 100% * Percentage increase = (r1 / r1) * 100% = 1 * 100% = 100%