One piece of copper jewelry at 104°C has exactly twice the mass of another piece, which is at 43.0°C. Both pieces are placed inside a calorimeter whose heat capacity is negligible. What is the final temperature inside the calorimeter (c of copper = 0.387 J/gK)?
To solve this problem, we can use the principle of conservation of energy. The total energy gained by the cooler copper piece will be equal to the total energy lost by the hotter copper piece.
The energy gained or lost by an object can be found using the equation:
Q = mcΔT
Where:
Q is the heat gained or lost by the object
m is the mass of the object
c is the specific heat capacity of the material
ΔT is the change in temperature
Let's calculate the energy gained by the cooler copper piece:
m1 = mass of the cooler copper piece
T1 = initial temperature of the cooler copper piece = 43.0°C
Tf = final temperature inside the calorimeter
c = specific heat capacity of copper = 0.387 J/gK
Q1 = m1c(Tf - T1)
Next, let's calculate the energy lost by the hotter copper piece:
m2 = mass of the hotter copper piece = 2m1 (twice the mass of the cooler copper piece)
T2 = initial temperature of the hotter copper piece = 104°C
Q2 = m2c(Tf - T2)
Since the total energy gained by the cooler copper piece is equal to the total energy lost by the hotter copper piece, we can equate the two equations:
m1c(Tf - T1) = m2c(Tf - T2)
Substituting m2 = 2m1:
m1c(Tf - T1) = (2m1)c(Tf - T2)
Simplifying:
Tf - T1 = 2(Tf - T2)
Tf - T1 = 2Tf - 2T2
Tf - Tf = -T1 + 2T2
T2 - T1 = Tf
Tf = T2 - T1
Now we can plug in the values and calculate the final temperature:
T2 = 104°C
T1 = 43.0°C
Tf = 104°C - 43.0°C
Tf = 61.0°C
Therefore, the final temperature inside the calorimeter is 61.0°C.