Question

Calculate the change in entropy that occurs when 17.68 g of ice at -12.7°C is placed in 54.05 g of water at 100.0°C in a perfectly insulated vessel. Assume that the molar heat capacities for H2O(s) and H2O(l) are 37.5 J K-1 mol-1 and 75.3 J K-1 mol-1, respectively, and the molar enthalpy of fusion for ice is 6.01 kJ/mol.

Answer 1

the heat capacities are given in units of moles and Kelvins, so you'll have to convert everything to these units

The weight of one mole of H2O = 1.008*2 + 15.999 = 18.015 g

17.68 g ice = 17.68g * (1 m / 18.015 g ) = .981 moles ice = .981 mole H20
54.05 g water = 3 moles liquid water

T(Kelvin) = T(Celsius) + 273
-12.7 C = 260.3 K
100 C = 373 K

First, the ice is brought to its melting point of 273 K:

37.5 J K-1 mol-1*.981 mol = 36.79 J K-1
The entropy change for this process is 36.79 ln (Tf - Ti)
where Tf is final temperature (273 K); Ti is initial temperature (260.3 K)

= 36.79 ln(12.7) = 93.5 J/K



Next, the ice melts:
.981 * 6.01 kJ/mol = 5.89 kJ
for a change of state at constant temperature the entropy change is

Q/T = 5.89 kJ / 273 K = 21.6 J/K

Next 17.68 g of ice at 273 K and 54.05 g water reach the same temperature; Using the first law of thermodynamics, deltaQ for this entire process is zero because the container is insulated. Summing up the heats of all the processes to 0:


36.79 J K-1 (273-260.3) + 5890 + .981 *75.3*(Tf-273) + 3*75.3*(Tf - 373) = 0

467.23 + 5890 + 73.87*(Tf-273) + 225.9*(Tf-373) = 0

6356 + 73.87*Tf - 20166 + 225.9*Tf - 84261 = 0

6356 +299.77*Tf - 104427 = 0

299.77*Tf = 98071
Tf = 327 K This is the final temperature of the mixture after it has reached equilibrium.

The change in entropy in bringing the melted ice to this temperature is

.981*75.3*ln(327-273) = 294.6 J/K

The change in entropy in bringing the liquid water down to this temperature is

3*75.3*ln(327-373) = an imaginary number

The total change in entropy is 93.5 J/K + 21.6 J/K + 294.6 J/K = 409.4 J/K