A 14.0-g sample of ice at -18.0°C is mixed with 122.0 g of water at 87.0°C. Calculate the final temperature of the mixture assuming no heat loss to the surroundings. The heat capacities of H2O(s) and H2O(l) are 2.08 and 4.18 J/g · °C, respectively, and the enthalpy of fusion for ice is 6.02 kJ/mol.
I would work this in two stages.
{mass ice x specific heat ice x [0-(-30)]} + [(mass warm water x specific heat water x (x-87)] = 0
That will take care of moving the ice T to zero C. x is the temperature of the mixture. The next step actually is three steps in one.
heat to melt ice at zero C + heat added to melted ice to final T + heat lost by water at higher T to Tfinal.
[mass ice x heat fusion] + [mass melted ice x speicif heat ice water x (Tfinal-Tinitial)] + [mass warm water at x temperature x specific heat water x (Tfinal-Tinitial)] = 0
Tinitial for the warm water will be x value determined in the first step.
Post your work if you get stuck.
To calculate the final temperature of the mixture, we need to apply the principle of conservation of energy. We assume that there is no heat loss to the surroundings, so the heat gained by the water will be equal to the heat lost by the ice.
To start, we can calculate the heat gained by the water using the formula:
Q_water = m_water * C_water * (T_final - T_initial)
where:
m_water is the mass of water (122.0 g)
C_water is the heat capacity of liquid water (4.18 J/g · °C)
T_initial is the initial temperature of water (87.0°C)
T_final is the final temperature of the mixture (unknown)
Now, let's calculate the heat lost by the ice. Since the ice is initially at -18.0°C and we assume it will reach its melting point (0°C), we need to consider two steps: heating the ice to 0°C and then melting the ice at 0°C.
Step 1: Heating the ice to 0°C
The heat lost during this step can be calculated using:
Q_ice_heating = m_ice * C_ice * (0 - T_initial_ice)
where:
m_ice is the mass of ice (14.0 g)
C_ice is the heat capacity of ice (2.08 J/g · °C)
T_initial_ice is the initial temperature of ice (-18.0°C)
Step 2: Melting the ice at 0°C
The heat lost during melting can be calculated using the enthalpy of fusion:
Q_ice_melting = m_ice * ΔH_fusion
where:
ΔH_fusion is the enthalpy of fusion for ice (6.02 kJ/mol)
Note that we need to convert the mass of ice to moles to use this value.
Next, we equate the heat lost by the ice to the heat gained by the water:
Q_ice_heating + Q_ice_melting = Q_water
Combining the above equations and solving for T_final, we can find the final temperature of the mixture.