Ice cubes at exactly 0°C with a total mass of 54.0 g are combined with 140 g of water at 75°C in an insulated container. Given that the heat of fusion (ΔH_fus) is 6.02 kJ/mol and the specific heat capacity of water (c_water) is 4.18 J/g°C, what is the final temperature of the system if no heat is lost to the surroundings?

• A. 25°C
• B. 50°C
• C. 15°C
• D. 0°C

Answer: (C)

Verified step by step guidance

1. Step 1: Begin by understanding the concept of heat transfer. In an insulated container, the heat lost by the warmer substance (water at 75°C) will be equal to the heat gained by the colder substance (ice at 0°C) until thermal equilibrium is reached.
2. Step 2: Calculate the amount of heat required to melt the ice. Use the formula: \( q = n \times \Delta H_{fus} \), where \( n \) is the number of moles of ice. First, convert the mass of ice to moles using the molar mass of water (18.02 g/mol).
3. Step 3: Calculate the heat lost by the water as it cools down. Use the formula: \( q = m \times c_{water} \times \Delta T \), where \( m \) is the mass of water, \( c_{water} \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. Assume the final temperature is \( T_f \).
4. Step 4: Set up the equation for heat balance: the heat gained by the ice (to melt and then warm up to \( T_f \)) should equal the heat lost by the water (cooling down to \( T_f \)). Solve for \( T_f \) using the equations from steps 2 and 3.
5. Step 5: Consider the phase change of ice to water and the subsequent warming of the melted ice. Calculate the heat required for the ice to reach the final temperature \( T_f \) using the specific heat capacity of water. Combine this with the heat of fusion calculated earlier to find the total heat absorbed by the ice.