A 37.0 g ice cube at -15.0 degreesC is placed in 224 g of water at 48.0 degreesC. Find the final temperature of the system when equilibrium is reached. Ignore the heat capacity of the container and assume this is in a calorimeter, i.e. the system is thermally insulated from the surroundings. Give your answer in degreesC with 3 significant figures.
Specific heat of ice: 2.090 J/g K
Specific heat of water: 4.186 J/g K
Latent heat of fusion for water: 333 J/g
To solve this problem, we can use the principle of conservation of energy. The energy lost by the hot water will be equal to the energy gained by the ice cube until equilibrium is reached.
First, let's calculate the heat lost by the water.
Q1 = mass of water * specific heat of water * change in temperature
Q1 = 224 g * 4.186 J/gK * (final temperature - 48.0°C)
Next, let's calculate the heat gained by the ice cube.
Q2 = mass of ice * latent heat of fusion + mass of ice * specific heat of ice * change in temperature
Q2 = 37.0 g * 333 J/g + 37.0 g * 2.090 J/gK * (final temperature - (-15.0°C))
Since the system is thermally insulated, the heat lost by the water is equal to the heat gained by the ice cube.
Q1 = Q2
We can equate the two expressions:
224 g * 4.186 J/gK * (final temperature - 48.0°C) = 37.0 g * 333 J/g + 37.0 g * 2.090 J/gK * (final temperature - (-15.0°C))
Simplifying the equation:
938.464 * (final temperature - 48.0°C) = 12351 + 77.23 * (final temperature + 15.0°C)
Expanding the equation:
938.464 * final temperature - 44934.592 = 12351 + 77.23 * final temperature + 1163.45
Rearranging the equation:
938.464 * final temperature - 77.23 * final temperature = 12351 + 1163.45 + 44934.592
Combining like terms:
861.234 * final temperature = 57349.042
Dividing both sides by 861.234:
final temperature = 66.678°C
Therefore, the final temperature of the system when equilibrium is reached is 66.678°C.
To find the final temperature of the system when equilibrium is reached, we can use the principle of conservation of energy. The energy gained by the water must be equal to the energy lost by the ice during the process.
Let's break down the process into different steps:
1. Heat absorbed by the water:
The water is at an initial temperature of 48.0 degreesC and has a mass of 224 g. The specific heat of water is 4.186 J/g K. Therefore, the heat absorbed by the water can be calculated using the formula:
Q_water = mass_water * specific_heat_water * change_in_temperature_water
Here, the change in temperature for the water is the difference between the final temperature and the initial temperature of the water.
2. Heat lost by the ice:
The ice is at an initial temperature of -15.0 degreesC and has a mass of 37.0 g. In addition to the energy required to raise the temperature of the ice to its melting point, the ice must also undergo a phase change from solid to liquid. The formula for the heat lost by the ice can be calculated as:
Q_ice = (mass_ice * specific_heat_ice * change_in_temperature_ice) + (mass_ice * latent_heat_fusion)
Here, the change in temperature for the ice is the difference between its initial temperature and its melting point.
3. Equating the heat absorbed by the water and the heat lost by the ice:
Since the system is thermally insulated, there is no heat exchange with the surroundings. Therefore, the heat absorbed by the water must be equal to the heat lost by the ice. We can set up the equation:
Q_water = Q_ice
By substituting the formulas for Q_water and Q_ice, we can solve for the final temperature (T_final).
Now, let's plug in the values:
mass_water = 224 g
mass_ice = 37.0 g
specific_heat_water = 4.186 J/g K
specific_heat_ice = 2.090 J/g K
latent_heat_fusion = 333 J/g
initial_temperature_water = 48.0 degreesC
initial_temperature_ice = -15.0 degreesC
First, calculate the change in temperature for the water and ice:
change_in_temperature_water = T_final - initial_temperature_water
change_in_temperature_ice = 0 degreesC (melting point)
Now, substitute the values into the equations:
Q_water = (224 g) * (4.186 J/g K) * (T_final - 48.0 degreesC)
Q_ice = (37.0 g) * (2.090 J/g K) * (0 degreesC) + (37.0 g) * (333 J/g)
Set Q_water equal to Q_ice and solve for T_final:
(224 g) * (4.186 J/g K) * (T_final - 48.0 degreesC) = (37.0 g) * (2.090 J/g K) * (0 degreesC) + (37.0 g) * (333 J/g)
Simplifying the equation:
(224 g) * (4.186 J/g K) * (T_final - 48.0 degreesC) = (37.0 g) * (333 J/g)
Dividing both sides by (224 g) * (4.186 J/g K) and solving for T_final:
T_final - 48.0 degreesC = [(37.0 g) * (333 J/g)] / [(224 g) * (4.186 J/g K)]
T_final - 48.0 degreesC = 5.006
T_final = 5.006 degreesC + 48.0 degreesC
T_final = 53.0 degreesC
Therefore, the final temperature of the system when equilibrium is reached is 53.0 degreesC.