The ceramic coffee cup in the following figure (Figure 1), with m = 114 g and c=1090J/(kg⋅K), is initially at room temperature (24.0 ∘C). If 225 g of 80.5 ∘C coffee and 12.4 g of 5.00 ∘C cream are added to the cup, what is the equilibrium temperature of the system? Assume that no thermal energy is exchanged with the surroundings, and that the specific heat of coffee and cream are the same as that of water.
To find the equilibrium temperature of the system, we need to understand the concept of heat transfer and use the principle of conservation of energy. The heat transferred to the coffee cup can be calculated by using the equation:
Q = mcΔT
Where:
Q is the heat transferred to the system
m is the mass of the substance (coffee or cream)
c is the specific heat capacity of the substance (assumed to be the same as that of water)
ΔT is the change in temperature
In this case, we have two substances: coffee and cream. The heat transferred to the coffee cup can be calculated separately for the coffee and the cream, and then combined to find the total heat transferred.
First, let's calculate the heat transferred due to the coffee. Using the given values:
m coffee = 225 g
c coffee = c water = 4.18 J/(g⋅K) (specific heat capacity of water)
ΔT coffee = final temperature - initial temperature = Tf - 80.5°C
Next, let's calculate the heat transferred due to the cream. Using the given values:
m cream = 12.4 g
c cream = c water = 4.18 J/(g⋅K) (specific heat capacity of water)
ΔT cream = final temperature - initial temperature = Tf - 5.00°C
The total heat transferred to the coffee cup can be found by summing the heat transferred due to the coffee and the cream.
Q total = Q coffee + Q cream
Q total = (m coffee * c coffee * ΔT coffee) + (m cream * c cream * ΔT cream)
Now, we can set the total heat transferred to the coffee cup equal to the heat capacity of the coffee cup, and solve for the final temperature.
Q total = mcΔT
Where:
m = 114 g (mass of the ceramic coffee cup)
c = 1090 J/(kg⋅K) (specific heat capacity of the ceramic coffee cup)
ΔT = Tf - 24.0°C (final temperature - initial temperature)
Substituting the values:
(m coffee * c coffee * ΔT coffee) + (m cream * c cream * ΔT cream) = mcΔT
Simplifying and solving for Tf:
(m coffee * c coffee * ΔT coffee) + (m cream * c cream * ΔT cream) = m * c * ΔT
Plug in the respective values and solve for Tf to find the equilibrium temperature of the system.