Lead pellets of total mass 0.46 kg are heated to 107 C and then placed in a well-insulated aluminum cup of mass 0.15 kg that contains 0.52 kg of water initially at 13.8 C. What is the equilibrium temperature of the mixture?
To find the equilibrium temperature of the mixture, we can use the principle of conservation of energy. The heat gained by the water and the cup must be equal to the heat lost by the lead pellets. We can express this mathematically using the equation:
(heat gained by water + heat gained by cup) = heat lost by lead pellets
The heat gained by the water can be calculated using the formula:
q water = m water * c water * ΔT
where m water is the mass of water, c water is the specific heat capacity of water, and ΔT is the change in temperature.
The heat gained by the cup can be calculated in a similar way:
q cup = m cup * c cup * ΔT
where m cup is the mass of the cup and c cup is the specific heat capacity of the cup.
The heat lost by the lead pellets can be calculated using:
q pellets = m pellets * c pellets * ΔT
where m pellets is the mass of the lead pellets, c pellets is the specific heat capacity of lead, and ΔT is the change in temperature.
Since the problem states that the mixture reaches equilibrium, the final temperature of the mixture will be the same. Therefore, we can rewrite the equation as:
m water * c water * ΔT + m cup * c cup * ΔT = m pellets * c pellets * ΔT
We can rearrange this equation to solve for ΔT:
ΔT = (m pellets * c pellets * ΔT) / (m water * c water + m cup * c cup)
Substituting the given values:
m pellets = 0.46 kg
c pellets (specific heat capacity of lead) = 128 J/kg°C
m water = 0.52 kg
c water (specific heat capacity of water) = 4186 J/kg°C
m cup = 0.15 kg
c cup (specific heat capacity of aluminum) = 900 J/kg°C
Plugging these values into the equation, we can find ΔT:
ΔT = (0.46 kg * 128 J/kg°C * ΔT) / ((0.52 kg * 4186 J/kg°C) + (0.15 kg * 900 J/kg°C))
Now we need to solve for ΔT.