Four ice cubes at exactly 0 degrees Celsius having a total mass of 55.0g are combined with 120g of water at 77 degrees Celsius in an insulated container.
If no heat is lost to the surroundings, what will be the final temperature of the mixture?
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To determine the final temperature of the mixture, we can use the principle of conservation of energy, specifically the law of conservation of heat.
The equation for heat transfer is given by:
Q = m * c * ΔT
Where:
Q = heat energy transferred (in joules)
m = mass of the substance (in grams)
c = specific heat capacity of the substance (in J/g°C)
ΔT = change in temperature (in °C)
In this case, we have two substances: the ice cubes and the water. Before they reach equilibrium, heat will be transferred from the water to the ice cubes until they reach the same final temperature.
Let's calculate the amount of heat transferred between the water and the ice cubes using this equation.
For the water:
Q1 = m1 * c1 * ΔT1
m1 = 120g (mass of water)
c1 = specific heat capacity of water = 4.18 J/g°C (approximation)
ΔT1 = final temperature - initial temperature = final temperature - 77°C
For the ice cubes:
Q2 = m2 * c2 * ΔT2
m2 = 55g (mass of ice cubes)
c2 = specific heat capacity of ice = 2.09 J/g°C (approximation)
ΔT2 = final temperature - initial temperature = final temperature - 0°C
Since no heat is lost to the surroundings, the amount of heat transferred from the water to the ice cubes (Q1) must be equal to the amount of heat gained by the ice cubes (Q2). Therefore:
Q1 = Q2
m1 * c1 * ΔT1 = m2 * c2 * ΔT2
Substituting the known values:
120 * 4.18 * (final temperature - 77) = 55 * 2.09 * final temperature
Simplifying the equation:
(499.2 * final temperature) - (498.96 * 77) = 114.95 * final temperature
Solving for final temperature:
499.2 * final temperature - 498.96 * 77 = 114.95 * final temperature
38419.2 = 615.75 * final temperature
final temperature = 38419.2 / 615.75
final temperature ≈ 62.42°C
Therefore, the final temperature of the mixture will be approximately 62.42°C.