Mass of ice, mice = 1 kg
Temperature of ice, Tice = 0 + 273 = 273 K
Mass of water, mwater = 12 kg
Temperature of water, Twater = 27 + 273 = 300 K
Surrounding temperature, T0 = 15 + 273 = 288 K
Specific heat of water = 4.18 kJ/kg K
Specific heat of ice = 2.1 kJ/kg K
Latent heat of ice = 333.5 kJ/kg
Let Tc = common temperature when heat flows between ice and water stops.
Heat lost by water = Heat gained by ice i.e.,
12 × 4.18(300 – Tc ) = 4.18(Tc – 273) + 333.5
or 15048 – 50.16Tc = 4.18Tc – 1141.14 + 333.5
or 54.34 Tc = 15855.64
∴ Tc = 291.8 K or 18.8°C
Change of entropy of water
= 12 × 4.18 loge \(\left(\cfrac{291.8}{300}\right)\) = – 1.39 kJ/K\(\cfrac{333.5}{273}\) =
Change of entropy of ice = 1 × 4.18 loge \(\left(\cfrac{291.8}{300}\right)\) = 1.499 kJ/K
Net change of entropy, ∆S = – 1.39 + 1.499 = 0.109 kJ/K
Hence, net increase in entropy = 0.109 kJ/K.
Increase in unavailable energy = T0∆S = 288 × 0.109 = 31.39 kJ.