Question

two 20.0 g ice cubes at (-13.0) °c are placed into 245 g of water at 25.0 °c. assuming no energy is transferred to or from the surroundings, calculate the final temperature, (t_f), of the water after all the ice melts.

heat capacity of (ce{h2o(s)})	37.7 j/(mol·k)
heat capacity of (ce{h2o(l)})	75.3 j/(mol·k)
enthalpy of fusion of (ce{h2o})	6.01 kj/mol

(t_f = square) °c

Explanation:

Step1: Calculate moles of ice/water

Molar mass of $\text{H}_2\text{O} = 18.015\ \text{g/mol}$
Moles of ice: $n_{\text{ice}} = \frac{2 \times 20.0\ \text{g}}{18.015\ \text{g/mol}} = 2.220\ \text{mol}$
Moles of liquid water: $n_{\text{water}} = \frac{245\ \text{g}}{18.015\ \text{g/mol}} = 13.60\ \text{mol}$

Step2: Define heat balance equation

Heat gained by ice = Heat lost by warm water
Heat gained by ice = heat to warm ice to $0^\circ\text{C}$ + heat to melt ice + heat to warm melted ice to $T_f$
Heat lost by warm water = heat to cool warm water to $T_f$

Step3: Write heat terms with formulas

1. Warm ice to $0^\circ\text{C}$: $q_1 = n_{\text{ice}} \times C_{s,\text{ice}} \times (0 - (-13.0))$
2. Melt ice: $q_2 = n_{\text{ice}} \times \Delta H_{\text{fusion}}$
3. Warm melted ice to $T_f$: $q_3 = n_{\text{ice}} \times C_{s,\text{liquid}} \times (T_f - 0)$
4. Cool warm water to $T_f$: $q_4 = n_{\text{water}} \times C_{s,\text{liquid}} \times (25.0 - T_f)$

Step4: Substitute values and solve

Convert $\Delta H_{\text{fusion}}$ to J/mol: $6.01\ \text{kJ/mol} = 6010\ \text{J/mol}$
$q_1 + q_2 + q_3 = q_4$
$2.220 \times 37.7 \times 13.0 + 2.220 \times 6010 + 2.220 \times 75.3 \times T_f = 13.60 \times 75.3 \times (25.0 - T_f)$

Calculate each term:
$q_1 = 2.220 \times 37.7 \times 13.0 = 1080\ \text{J}$
$q_2 = 2.220 \times 6010 = 13340\ \text{J}$
Left side total: $1080 + 13340 + 167.2 T_f = 14420 + 167.2 T_f$
Right side: $13.60 \times 75.3 \times 25.0 - 13.60 \times 75.3 T_f = 25600 - 1024 T_f$

Rearrange to solve for $T_f$:
$167.2 T_f + 1024 T_f = 25600 - 14420$
$1191.2 T_f = 11180$
$T_f = \frac{11180}{1191.2}$

Answer:

$9.39^\circ\text{C}$