one piece of copper metal at 105 C has twice the mass of another copper piece at 45 C. What is the final temperature if these two pieces are placed in a calorimeter. Specific heat of copper is 0.387 J/g.K.

To solve this problem, we can apply the principle of conservation of energy. The heat lost by the hot copper piece will be equal to the heat gained by the cold copper piece and the calorimeter.

Let's solve it step by step:

Step 1: Identify the given information.
- Mass of the first copper piece (hot): m1 = 2m
- Mass of the second copper piece (cold): m2 = m
- Initial temperature of the first copper piece: T1 = 105 °C
- Initial temperature of the second copper piece: T2 = 45 °C
- Specific heat of copper: c = 0.387 J/g.K

Step 2: Calculate the heat lost by the hot copper piece.
The heat lost (Q1) is given by the equation:
Q1 = m1 * c * (Tf - T1)
where Tf is the final temperature that we need to calculate.

Step 3: Calculate the heat gained by the cold copper piece and the calorimeter.
The heat gained (Q2) is given by the equation:
Q2 = (m2 + Calorimeter mass) * c * (Tf - T2)
where Calorimeter mass represents the mass of the calorimeter.

Step 4: Equate the two equations to solve for Tf.
Since the heat lost is equal to the heat gained, we can set Q1 equal to Q2:
m1 * c * (Tf - T1) = (m2 + Calorimeter mass) * c * (Tf - T2)

Step 5: Solve for Tf.
Rearrange the equation to isolate Tf:
m1 * (Tf - T1) = (m2 + Calorimeter mass) * (Tf - T2)
m1 * Tf - m1 * T1 = m2 * Tf - m2 * T2 + Calorimeter mass * Tf - Calorimeter mass * T2
(m1 - m2) * Tf = m1 * T1 - m2 * T2 + Calorimeter mass * (T2 - Tf)
Tf = (m1 * T1 - m2 * T2 + Calorimeter mass * (T2 - Tf)) / (m1 - m2 + Calorimeter mass)

Step 6: Substitute the given values and solve.
Substitute the values into the equation and solve for Tf:
Tf = (2m * 105 - m * 45 + Calorimeter mass * (45 - Tf)) / (2m - m + Calorimeter mass)

Note that the calorimeter mass is not given in the problem statement. Make sure to substitute it with the actual mass of the calorimeter.

Simplifying this equation further is not possible without knowing the value of Calorimeter mass.

Once you have the actual mass of the calorimeter, you can substitute it and solve the equation to find the final temperature (Tf).

i figured this one out