A copper calorimeter can with mass .100 kg contains .160 kg of water and .018 kg of ice in thermal equilibrium at atmospheric pressure. If .750 kg of lead at a temperature of 255 degrees Celsius is dropped into the can what is the final temperature of the system if no heat is lost to the surroundings?
To determine the final temperature of the system, we can apply the principle of conservation of energy.
First, let's find the amount of heat gained or lost by each component of the system.
1. Heat gained or lost by the copper calorimeter:
The heat gained or lost by the copper calorimeter (Qc) can be calculated using the formula:
Qc = mc * cc * ΔTc
where:
- mc is the mass of the copper calorimeter (0.100 kg)
- cc is the specific heat capacity of copper (approximately 387 J/kg⋅°C)
- ΔTc is the change in temperature of the calorimeter
Since we assume no heat is lost to the surroundings, the heat gained by the copper calorimeter will be equal to the heat lost by the lead and the heat lost or gained by the ice-water mixture.
2. Heat gained or lost by the lead:
The heat gained or lost by the lead (Ql) can be calculated using the formula:
Ql = ml * cl * ΔTl
where:
- ml is the mass of the lead (0.750 kg)
- cl is the specific heat capacity of lead (approximately 129 J/kg⋅°C)
- ΔTl is the change in temperature of the lead
3. Heat gained or lost by the ice-water mixture:
The heat gained or lost by the ice-water mixture (Qi) can be calculated using the formula:
Qi = mi * ci * ΔTi
where:
- mi is the total mass of the ice and water (0.160 kg)
- ci is the specific heat capacity of water (approximately 4186 J/kg⋅°C)
- ΔTi is the change in temperature of the ice-water mixture
Since the system is in thermal equilibrium, the change in temperature of the ice-water mixture will determine the final temperature of the system.
Now, let's calculate the temperatures changes for each component:
ΔTc = ΔTl = ΔTi = Tf - Ti
where:
- Tf is the final temperature of the system
- Ti is the initial temperature of the system
Since the system is in thermal equilibrium, the initial temperature of the system is the same for all components. We can assume it is the temperature of the ice-water mixture (0 °C).
Now, let's substitute the known values into the equations and solve for Tf:
Qc + Ql + Qi = 0
(mc * cc * ΔTc) + (ml * cl * ΔTl) + (mi * ci * ΔTi) = 0
(0.100 kg * 387 J/kg⋅°C * ΔTc) + (0.750 kg * 129 J/kg⋅°C * ΔTl) + (0.160 kg * 4186 J/kg⋅°C * ΔTi) = 0
Solve the equation to find the values for ΔTl and ΔTi:
(0.100 kg * 387 J/kg⋅°C * ΔTc) + (0.750 kg * 129 J/kg⋅°C * ΔTl) + (0.160 kg * 4186 J/kg⋅°C * ΔTi) = 0
After obtaining the values for ΔTl and ΔTi, substitute them back into the equation:
Tf = Ti + ΔTl = Ti + ΔTi
Now, solve for Tf.