Suppose 11.0 g of ice at -10.0∘C is placed into 300.0 g of water in a 200.0-g copper calorimeter. The final temperature of the water and copper calorimeter is 18.0∘C.
1)What was the initial common temperature of the water and copper? (Express your answer to three significant figures.)
To find the initial common temperature of the water and copper, we need to use the principle of conservation of energy and the concept of heat transfer.
First, we need to calculate the heat gained by the water and the copper calorimeter. The heat gained by the water can be expressed as:
q_water = m_water * C_water * ΔT_water
where:
- q_water is the heat gained by the water
- m_water is the mass of the water (300.0 g)
- C_water is the specific heat capacity of water (4.18 J/g°C)
- ΔT_water is the change in temperature of the water
Since we know the final temperature of the water (18.0°C) and want to find the initial common temperature, we can express the change in temperature as:
ΔT_water = T_final - T_initial
Next, we calculate the heat gained by the copper calorimeter. The heat gained by the copper calorimeter can be expressed as:
q_copper = m_copper * C_copper * ΔT_copper
where:
- q_copper is the heat gained by the copper calorimeter
- m_copper is the mass of the copper calorimeter (200.0 g)
- C_copper is the specific heat capacity of copper (0.385 J/g°C)
- ΔT_copper is the change in temperature of the copper calorimeter
Similar to the water, since we know the final temperature of the copper and want to find the initial common temperature, we can express the change in temperature as:
ΔT_copper = T_final - T_initial
According to the principle of conservation of energy, the heat lost by the ice is equal to the heat gained by the water and copper calorimeter:
q_ice = q_water + q_copper
Now, we can calculate the initial common temperature (T_initial) by rearranging the above equation:
T_initial = T_final - (q_ice / (m_water * C_water + m_copper * C_copper))
To find q_ice, we need to calculate the heat lost by the ice. The heat lost by the ice can be expressed as:
q_ice = m_ice * C_ice * ΔT_ice
where:
- q_ice is the heat lost by the ice
- m_ice is the mass of the ice (11.0 g)
- C_ice is the specific heat capacity of ice (2.09 J/g°C)
- ΔT_ice is the change in temperature of the ice
Since the ice is initially at -10.0°C and we want to find the initial common temperature, we can express the change in temperature as:
ΔT_ice = T_initial - T_ice
Now, we substitute the values into the equations and solve for T_initial:
ΔT_ice = T_initial - T_ice
ΔT_ice = T_initial - (-10.0°C)
T_initial = ΔT_ice + (-10.0°C)
T_initial = [m_ice * C_ice * (T_initial - (-10.0°C))] / (m_water * C_water + m_copper * C_copper)
Now, we plug in the known values:
T_initial = [(11.0 g) * (2.09 J/g°C) * (T_initial - (-10.0°C))] / [(300.0 g) * (4.18 J/g°C) + (200.0 g) * (0.385 J/g°C)]
This equation can be solved by substituting in different values for T_initial and iteratively finding the correct value that satisfies the equation.