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 find the final temperature of the system when equilibrium is reached, we need to consider the heat gained or lost by the ice and the heat gained or lost by the water. Let's go step by step.

Step 1: Calculate the heat gained or lost by the ice during the heating process (from -15.0 degreesC to 0 degreesC):

The heat gained or lost by the ice can be calculated using the formula: Q = m * C * ΔT, where Q is the heat gained or lost, m is the mass, C is the specific heat, and ΔT is the change in temperature.

In this case, the mass of the ice is 37.0 g and the specific heat of ice is 2.090 J/g K. The change in temperature is from -15.0 degreesC to 0 degreesC, so ΔT = 0 - (-15.0) = 15.0 degreesC.

Therefore, the heat gained or lost by the ice is Q = 37.0 g * 2.090 J/g K * 15.0 degreesC = 1,239.15 J.

Step 2: Calculate the heat gained or lost by the water during the cooling process (from 48.0 degreesC to 0 degreesC):

The heat gained or lost by the water can be calculated using the same formula: Q = m * C * ΔT.

In this case, the mass of the water is 224 g and the specific heat of water is 4.186 J/g K. The change in temperature is from 48.0 degreesC to 0 degreesC, so ΔT = 0 - 48.0 = -48.0 degreesC.

Therefore, the heat gained or lost by the water is Q = 224 g * 4.186 J/g K * -48.0 degreesC = -39,767.04 J.

Step 3: Calculate the heat gained or lost during the phase change of the ice to water at 0 degreesC:

The heat gained or lost during the phase change (melting) of the ice can be calculated using the formula: Q = m * L, where Q is the heat gained or lost, m is the mass, and L is the latent heat of fusion.

In this case, the mass of the ice is 37.0 g and the latent heat of fusion for water is 333 J/g.

Therefore, the heat gained or lost during the phase change is Q = 37.0 g * 333 J/g = 12,321 J.

Step 4: Calculate the total heat gained or lost by the system:

The total heat gained or lost by the system is the sum of the heat gained or lost by the ice, heat gained or lost by the water, and the heat gained or lost during phase change:

ΔQ = 1,239.15 J + (-39,767.04 J) + 12,321 J
ΔQ = -26,206.89 J

Step 5: Use the principle of energy conservation:

In a thermally insulated system, the heat lost by one component is equal to the heat gained by the other component when they reach equilibrium.

Therefore, -ΔQ = ΔQ_water

-26,206.89 J = m_water * C_water * ΔT

Substituting the known values for mass and specific heat of water, and solving for ΔT:

-26,206.89 J = 224 g * 4.186 J/g K * ΔT

ΔT = -26,206.89 J / (224 g * 4.186 J/g K)
ΔT = -29.673 K

Step 6: Calculate the final temperature:

The final temperature is the initial temperature of the water (48.0 degreesC) minus the change in temperature:

Final temperature = 48.0 degreesC - 29.673 K
Final temperature = 18.327 degreesC

Therefore, when equilibrium is reached, the final temperature of the system is approximately 18.327 degreesC.

To find the final temperature of the system when equilibrium is reached, we need to calculate the amount of heat transferred between the ice cube and the water.

First, let's calculate the heat absorbed by the ice cube to reach 0 degrees Celsius:

Q1 = mass of ice cube * specific heat of ice * temperature change
= 37.0 g * 2.090 J/g K * (0 - (-15.0) degrees Celsius)
= 37.0 g * 2.090 J/g K * 15.0 degrees Celsius
= 1152.45 J

Next, let's calculate the heat absorbed by the ice cube during the phase change from solid to liquid (fusion):

Q2 = mass of ice cube * latent heat of fusion for water
= 37.0 g * 333 J/g
= 12321 J

Now, let's calculate the heat absorbed by the water to reach the final temperature:

Q3 = mass of water * specific heat of water * temperature change
= 224 g * 4.186 J/g K * (final temperature - 48.0 degrees Celsius)

Since the system is thermally insulated, the heat absorbed by the ice cube must be equal to the heat absorbed by the water:

Q1 + Q2 = Q3

1152.45 J + 12321 J = 224 g * 4.186 J/g K * (final temperature - 48.0 degrees Celsius)

1354738 J = 939.264 g * (final temperature - 48.0 degrees Celsius)

(final temperature - 48.0 degrees Celsius) = 1354738 J / (939.264 g * 4.186 J/g K)

(final temperature - 48.0 degrees Celsius) = 324.124 K

final temperature = 324.124 K + 48.0 degrees Celsius

final temperature ≈ 372.12 degrees Celsius

Therefore, the final temperature of the system when equilibrium is reached is approximately 372 degrees Celsius.