Specific Heat Capacities (c)
Ice: 2108 Joules Per Kilogram-Dot-Kelvin
Water: 4187 Joules Per Kilogram-Dot-Kelvin
Steam: 1996 Joules Per Kilogram-Dot-Kelvin
Degrees Kelvin is found by Degrees Centigrade + 273.
The final temperature of this union of ice and water is found by the equation
micecice(Tfinal − Tice) = mwatercwater(Twater − Tfinal) or
(0.06247 kg)(2108 J/kg•K)(Tfinal − 273K) = (0.497 kg)(4187 J/kg•K)(283K − Tfinal).
Rewrite as 0.06328237397(Tfinal − 273K) = (283K − Tfinal) and obtain Tfinal equal to 282.4048397K
(or 9.4048397°C).
Replace the irreversible process (the actual one) with a reversible process in which the water in the cup is
successively placed in contact with an infinitesimally cooler reservoir and thereby cooled in an infinite number of tiny steps. A similar procedure is followed with the ice. The desired state of 9.4048397°C is then reached.
For masses 0.06247 kg & 0.497 kg with respective specific heat capacities of 2108 Joules Per Kilogram-Dot-Kelvin and 4187 Joules Per Kilogram-Dot-Kelvin, ΔS, the change in entropy, then equals
∫1dQ/T + ∫2dQ/T = micecice∫(from T-ice to T-final)(dT/T) + mwatercwater∫(from T-water to T-final)(dT/T).
ΔS then equals miceciceln(Tfinal/Tice) + mwatercwaterln(Tfinal/Twater). This is the entropy change for mixing, where mass mice at Tice is mixed with mass mwater at Twater, with a final temperature of Tfinal.
Placing values, ΔS = (0.06247 kg)(2108 J/kg•K)ln (282.4048397/273) +
(0.497 kg)(4187 J/kg•K)ln (282.4048397/283) which simplifies to 0.07930451528 J/K (Joules per Degree K).
