Ice in warm water: does it all melt?

Drop ice into warm water: an energy ledger checks whether the water can melt all the ice; final state is either water above 0 °C or an ice-water mix at 0 °C.

Find the largest ice mass that 300 g of water at 50 °C can melt completely (≈188 g at 0 °C ice).

Assuming all the ice melts and solving for a negative final temperature.

• A system is isolated when it exchanges no heat with its surroundings. When bodies at different temperatures are placed in contact inside such a system, heat flows from the hot ones to the cold ones until they share one temperature.
• Principle of calorimetry: in an isolated system, heat lost by the hot bodies = heat gained by the cold bodies.
• A calorimeter is the device for measuring heat. It is a metal vessel with a stirrer of the same metal, usually copper or aluminium, placed inside a wooden jacket with glass wool between them to keep heat from escaping.
• A thermometer passes through the lid into the liquid. The metal vessel itself takes part in the heat exchange, so its heat must be counted as well as the liquid's.
• Heat changes are found from Q = m s ΔT for each body. For mixtures the unknown is usually a specific heat or the final temperature.
• Worked example: an aluminium sphere of 0.047 kg at 100 °C is dropped into a copper calorimeter of 0.14 kg holding 0.25 kg of water at 20 °C; the final temperature is 23 °C.
• Heat gained by water and calorimeter = (0.25 × 4.18 × 10³ + 0.14 × 0.386 × 10³) × 3 = 3297.1 J. Heat lost by the sphere = 0.047 × s_Al × 77. Equating gives s_Al ≈ 911 J kg⁻¹ K⁻¹ = 0.911 kJ kg⁻¹ K⁻¹.
• Losses to the surroundings make a real measurement less exact, which is why the insulation, lid and quick readings matter.

Teaching this idea? Put the simulation on your own page or LMS with this code.