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?

Heat lost by lead = heat gained by water.

Write that with T(final) as the only unknown, using masses and specific heats, and solve for T(final), the equilibrium temperature.

You will need to look up the lead specific heat. I am sure you know the value for water.

None of the water will boil away in this case

Can you help me solve the problem bc I tried to do it as you said but I'm getting the wrong answer.

To find the equilibrium temperature of the mixture, we can use the principle of conservation of energy.

First, let's calculate the heat gained by the aluminum cup and the water when the lead pellets are placed in them. We'll assume that the specific heat capacity of the lead pellets, aluminum cup, and water are constant.

1. Calculate the heat gained by the aluminum cup:
The heat gained by the aluminum cup can be calculated using the formula:
Q = mcΔT

Here,
m = mass of the aluminum cup = 0.15 kg
c = specific heat capacity of aluminum = 900 J/kg°C (approximate value)
ΔT = change in temperature of aluminum cup

The change in temperature can be calculated using:
ΔT = Tf - Ti,
where Tf is the final temperature and Ti is the initial temperature.

In this case, the initial temperature of the aluminum cup is 13.8°C, and the final temperature will be the equilibrium temperature.

2. Calculate the heat gained by the water:
The heat gained by the water can be calculated using the formula:
Q = mwchwΔT

Here,
mw = mass of water = 0.52 kg
chw = specific heat capacity of water = 4186 J/kg°C (approximate value)
ΔT = change in temperature of water

Again, ΔT = Tf - Ti,
where Tf is the final temperature and Ti is the initial temperature.

In this case, the initial temperature of the water is 13.8°C, and the final temperature will be the equilibrium temperature.

3. Calculate the heat lost by the lead pellets:
The heat lost by the lead pellets can be calculated using the formula:
Q = mplcpΔT

Here,
mp = mass of lead pellets = 0.46 kg
lcp = specific heat capacity of lead pellets = 126 J/kg°C (approximate value)
ΔT = change in temperature of lead pellets

Since the lead pellets are initially at 107°C and the final temperature is the equilibrium temperature, ΔT = Tf - 107.

Now, since energy is conserved in the system, the heat lost by the lead pellets is equal to the heat gained by the aluminum cup and the water.

mplcpΔT = mcΔT + mwchwΔT

Let's substitute the given values and solve for Tf (equilibrium temperature):

0.46 kg * 126 J/kg°C * (Tf - 107) = 0.15 kg * 900 J/kg°C * (Tf - 13.8) + 0.52 kg * 4186 J/kg°C * (Tf - 13.8)

Simplify and solve the equation to find the value of Tf (equilibrium temperature).

Note: You can use algebraic techniques or numerical methods to solve this equation, such as substitution or iteration.

Once you've solved the equation, the resulting value of Tf will be the equilibrium temperature of the mixture.