A student mixes 85.1 g of water at 69.6°C with 58.1 g of water at 22.0°C in an insulated flask. What is the final temperature of the combined water?
heat lost by warm water + heat gained by cool water = 0
[mass warm H2O x specific heat H2O x (Tfinal-Tinnitial)] + [mass cool water x specific heat H2O x (Tfinal-Tinitial)] = 0. Solve for Tfinal
To find the final temperature of the combined water, we need to use the principle of energy conservation. The total heat gained by the cold water should be equal to the total heat lost by the hot water.
We can calculate the heat gained or lost using the formula:
Q = m * c * ΔT
Where:
Q = Heat gained or lost
m = Mass of the water
c = Specific heat capacity of water
ΔT = Change in temperature
Let's calculate the heat gained by the cold water first:
Q1 = m1 * c * ΔT1
Where:
m1 = Mass of cold water
ΔT1 = Change in temperature of cold water = Final temperature - Initial temperature
Next, let's calculate the heat lost by the hot water:
Q2 = m2 * c * ΔT2
Where:
m2 = Mass of hot water
ΔT2 = Change in temperature of hot water = Initial temperature - Final temperature
Since the total heat gained by the cold water is equal to the total heat lost by the hot water, we can set up the equation:
Q1 = Q2
m1 * c * ΔT1 = m2 * c * ΔT2
Plugging in the given values:
m1 = 85.1 g
m2 = 58.1 g
c = Specific heat capacity of water = 4.18 J/g°C
ΔT1 = Final temperature - 22.0°C
ΔT2 = 69.6°C - Final temperature
Now, we can solve the equation to find the final temperature:
m1 * c * (Final temperature - 22.0°C) = m2 * c * (69.6°C - Final temperature)
85.1 g * 4.18 J/g°C * (Final temperature - 22.0°C) = 58.1 g * 4.18 J/g°C * (69.6°C - Final temperature)
Simplifying the equation:
355.418 J * (Final temperature - 22.0°C) = 242.078 J * (69.6°C - Final temperature)
Expanding the equation:
355.418 J * Final temperature - 7819.196 J = 16833.8624 J - 242.078 J * Final temperature
Rearranging the equation:
355.418 J * Final temperature + 242.078 J * Final temperature = 16833.8624 J + 7819.196 J
597.496 J * Final temperature = 24653.0584 J
Dividing both sides by 597.496 J:
Final temperature = 41.2857°C
Therefore, the final temperature of the combined water is approximately 41.29°C.