A combination of .403 kg of water at 45.1 C, .471 kg of aluminum at 41.1 C, and .307 kg of copper at 50.4 C is mixed in an insulated container and allowed to come to thermal equilibrium. Determine the final temperature of the mixture. Neglect any energy transfer to or from the container and assume the specific heat of copper is 387 J/kg C and the specific heat of water is 4186 J/kg C. Answer in units of C.
To find the final temperature of the mixture, we can use the principle of conservation of energy.
The first step is to calculate the energy gained or lost by each substance when they reach thermal equilibrium.
For water:
The specific heat of water is 4186 J/kg C, and the initial temperature is 45.1 C.
The energy gained or lost by water can be calculated using the formula:
Q_water = (mass_water) x (specific heat_water) x (final temperature - initial temperature)
For aluminum:
The specific heat of aluminum is not given, but we can assume it to be 900 J/kg C (typical value for aluminum).
The initial temperature is 41.1 C.
The energy gained or lost by aluminum can be calculated using the formula:
Q_aluminum = (mass_aluminum) x (specific heat_aluminum) x (final temperature - initial temperature)
For copper:
The specific heat of copper is given as 387 J/kg C, and the initial temperature is 50.4 C.
The energy gained or lost by copper can be calculated using the formula:
Q_copper = (mass_copper) x (specific heat_copper) x (final temperature - initial temperature)
Since the container is insulated and there is no energy transfer to or from the container, the total energy gained by one substance must be equal to the total energy lost by the other substances. Therefore, we can set up the following equation:
Q_water + Q_aluminum + Q_copper = 0
Substituting the given values into the equation, we get:
(mass_water) x (specific heat_water) x (final temperature - initial temperature) + (mass_aluminum) x (specific heat_aluminum) x (final temperature - initial temperature) + (mass_copper) x (specific heat_copper) x (final temperature - initial temperature) = 0
Now, we can substitute the values into the equation and solve for the final temperature.
(0.403 kg) x (4186 J/kg C) x (final temperature - 45.1 C) + (0.471 kg) x (900 J/kg C) x (final temperature - 41.1 C) + (0.307 kg) x (387 J/kg C) x (final temperature - 50.4 C) = 0
Simplifying the equation and solving for the final temperature, we get:
(1704.158 J/C) x (final temperature) - (75,102.7264 J) = 0
Final temperature = (75,102.7264 J) / (1704.158 J/C)
Final temperature ≈ 44.07°C
Therefore, the final temperature of the mixture is approximately 44.07°C.