what is the minimum mass of ice at 0.0C that must be added to 1.00kg of water to cool the water from 25.0C to 12.0C. the heat of fusion is 333J/g; specific heat capacities of ice = 2.06J/gK; the heat capacity of water is 4.184J/gK.
I have no clue where to start.
To solve this problem, you can use the principle of conservation of energy. The heat lost by the water will be equal to the heat gained by the ice.
First, calculate the heat lost by the water:
Q_water = mass_water * specific_heat_water * change_in_temperature
mass_water = 1.00 kg (given)
specific_heat_water = 4.184 J/gK (given)
change_in_temperature = (25.0 - 12.0) °C = 13.0 °C
Convert kilograms to grams:
mass_water = 1.00 kg * 1000 g/kg = 1000 g
Q_water = 1000 g * 4.184 J/gK * 13.0 °C = 54,232 J
Next, calculate the heat gained by the ice:
Q_ice = mass_ice * heat_of_fusion + mass_ice * specific_heat_ice * change_in_temperature
heat_of_fusion = 333 J/g (given)
specific_heat_ice = 2.06 J/gK (given)
change_in_temperature = (0.0 - (-12.0)) °C = 12.0 °C
To find the mass of the ice, we need to determine how much heat is needed to cool the water to 0°C and then freeze it.
Q_ice = mass_ice * heat_of_fusion + mass_ice * specific_heat_ice * change_in_temperature
Rearranging the equation,
mass_ice = Q_ice / (heat_of_fusion + specific_heat_ice * change_in_temperature)
Substituting the given values,
mass_ice = 54232 J / (333 J/g + 2.06 J/gK * 12.0 °C)
Now, let's calculate it step by step:
1. Calculate the product of specific heat and the change in temperature:
specific_heat_ice * change_in_temperature = 2.06 J/gK * 12.0 °C
2. Calculate the sum of heat of fusion and the product from step 1:
heat_of_fusion + (specific_heat_ice * change_in_temperature) = 333 J/g + 2.06 J/gK * 12.0 °C
3. Calculate the mass of the ice:
mass_ice = 54232 J / (heat_of_fusion + (specific_heat_ice * change_in_temperature))
Now, you can substitute in the values and calculate:
mass_ice = 54232 J / (333 J/g + 2.06 J/gK * 12.0 °C)
Once you perform the calculation, you will find the minimum mass of ice required to cool the water.