Milk with a mass of 0.021 kg and a tempera-
ture of 13�C is added to 0.12 kg of coffee at
94�C.
What is the final temperature? Assume the
specific heat capacities of the two liquids are
the same as water, and disregard any energy
transfer to the liquids’ surroundings.
Answer in units of �C
To find the final temperature when milk and coffee are mixed, you can use the principle of conservation of energy. The total energy before mixing is equal to the total energy after mixing.
The equation for the transfer of heat energy is:
Q = mcΔT
Where:
Q = heat energy transferred
m = mass of the substance
c = specific heat capacity
ΔT = change in temperature
First, let's calculate the heat energy transferred from the milk:
Qmilk = mcΔTmilk
Given:
mass of milk (milk) = 0.021 kg
initial temperature of milk (Tmilk_initial) = 13°C
specific heat capacity of water (cmilk) = specific heat capacity of water = 4186 J/kg·°C
ΔTmilk = Tfinal - Tmilk_initial
Next, let's calculate the heat energy transferred from the coffee:
Qcoffee = mcΔTcoffee
Given:
mass of coffee (coffee) = 0.12 kg
initial temperature of coffee (Tcoffee_initial) = 94°C
specific heat capacity of water (ccoffee) = specific heat capacity of water = 4186 J/kg·°C
ΔTcoffee = Tfinal - Tcoffee_initial
Since the total energy is conserved:
Qmilk + Qcoffee = 0
Substituting the formulas:
mcΔTmilk + mcΔTcoffee = 0
Simplifying:
m(cΔTmilk + cΔTcoffee) = 0
Dividing by the mass:
cΔTmilk + cΔTcoffee = 0
Substituting the values:
(0.021 kg)(4186 J/kg·°C)(Tfinal - 13°C) + (0.12 kg)(4186 J/kg·°C)(Tfinal - 94°C) = 0
Expanding:
(87.606 J/°C)(Tfinal - 13°C) + (502.32 J/°C)(Tfinal - 94°C) = 0
Now, let’s rearrange the equation to solve for Tfinal:
(87.606 J/°C)(Tfinal) - (1139.878 J/°C) + (502.32 J/°C)(Tfinal) - (47290.08 J/°C) = 0
Combining like terms:
(589.926 J/°C)(Tfinal) - (48430.958 J/°C) = 0
Adding (48430.958 J/°C) to both sides:
(589.926 J/°C)(Tfinal) = (48430.958 J/°C)
Now, divide both sides by (589.926 J/°C):
Tfinal = (48430.958 J/°C) / (589.926 J/°C)
Solving for Tfinal:
Tfinal ≈ 82.16°C
Therefore, the final temperature is approximately 82.16°C.