A 44.0-g block of ice at -15.0°C is dropped into a calorimeter (of negligible heat capacity) containing of water at 5.0°C. When equilibrium is reached, how much of the ice will have melted? The specific heat of ice is 2090 J/kg ∙ K, that of water is 4186 J/kg ∙ K, and the latent heat of fusion of water is 33.5 × 104 J/kg
To determine how much of the ice will have melted, we need to calculate the heat exchanged between the ice and the water.
Step 1: Calculate the heat absorbed by the ice to reach 0°C (melting temperature).
The formula to calculate the heat absorbed is: Q = m * c * ΔT, where Q is the heat absorbed, m is the mass, c is the specific heat, and ΔT is the change in temperature.
Given:
Mass of ice (m) = 44.0 g = 0.044 kg
Specific heat of ice (c) = 2090 J/kg ∙ K
Initial temperature of ice (T1) = -15.0°C
Temperature change (ΔT) = 0°C - (-15.0°C) = 15.0°C
Q1 = m * c * ΔT
Q1 = 0.044 kg * 2090 J/kg ∙ K * 15.0 K
Q1 = 1383.6 J
Step 2: Calculate the heat absorbed to melt the ice at 0°C.
The formula to calculate the heat absorbed for a phase change is: Q = m * L, where Q is the heat absorbed, m is the mass, and L is the latent heat of fusion.
Given:
Latent heat of fusion (L) = 33.5 × 10^4 J/kg
Q2 = m * L
Q2 = 0.044 kg * (33.5 × 10^4 J/kg)
Q2 = 14740 J
Step 3: Calculate the heat absorbed by the water to reach the final temperature.
The formula to calculate the heat absorbed is: Q = m * c * ΔT, where Q is the heat absorbed, m is the mass, c is the specific heat, and ΔT is the change in temperature.
Given:
Mass of water (m) = ?
Since the mass of the water is not given, we can assume it is the same as the mass of the melted ice.
Mass of melted ice = 0.044 kg
Initial temperature of water (T1) = 5.0°C
Final temperature (T2) = 0°C
ΔT = T2 - T1
ΔT = 0°C - 5.0°C
ΔT = -5.0 K (Note: We use Kelvin for temperature differences)
Using the specific heat of water (c = 4186 J/kg ∙ K), we can calculate the heat absorbed by the water:
Q3 = m * c * ΔT
Q3 = 0.044 kg * 4186 J/kg ∙ K * (-5.0 K)
Q3 = -920.96 J (Note: Negative value indicates heat is released)
Step 4: Calculate the total heat exchanged.
The total heat exchanged is the sum of the three heats calculated: Q_total = Q1 + Q2 + Q3.
Q_total = 1383.6 J + 14740 J - 920.96 J
Q_total = 149202.64 J
Step 5: Calculate the mass of ice melted.
The mass of ice melted (m_melted) can be calculated using the formula: Q_melted = m_melted * L.
Q_melted = 149202.64 J
L = 33.5 × 10^4 J/kg
Q_melted = m_melted * L
m_melted = Q_melted / L
m_melted = 149202.64 J / (33.5 × 10^4 J/kg)
m_melted = 0.445 kg
Therefore, approximately 0.445 kg (or 445 g) of the ice will have melted.