Eight identical water drops, each of radius 0.50 mm, coalesce completely into one drop. Assume that all released surface energy is converted into heat in the water. Take S = 0.070 N/m, ρ = 1000 kg/m³ and specific heat c = 4200 J/(kg·K). If the temperature rise is N × 10⁻⁵ K, find N.
Physics · Properties of solid and fluid · properties of fluid and surface tension
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Question

Eight identical water drops, each of radius 0.50 mm, coalesce completely into one drop. Assume that all released surface energy is converted into heat in the water. Take S = 0.070 N/m, ρ = 1000 kg/m³ and specific heat c = 4200 J/(kg·K). If the temperature rise is N × 10⁻⁵ K, find N.

Physics · Properties of solid and fluid · properties of fluid and surface tension

G1 Physics · Accepted Answer

For eight equal drops, R = 2r. Initial total area: A(initial) = 8(4πr²) = 32πr². Final area: A(final) = 4π(2r)² = 16πr². Decrease in surface area: ΔA = 16πr². Released surface energy: E = SΔA = 16πSr². Total mass of the eight drops: m = 8[(4πr³/3)ρ] = [32πρr³]/3. If all released energy becomes heat, mcΔT = E. Therefore, ΔT = E/(mc) = [16πSr²] / {[32πρr³/3]c} = 3S/[2ρcr]. Substitute S = 0.070, ρ = 1000, c = 4200, r = 0.50 mm = 5×10⁻⁴ m. Thus, ΔT = 5×10⁻⁵ K. Hence, N = 5.