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------------------------------------------------ A 116 g cube of ice at 0 ∘C is dropped into 1.0 kg of water that was originally at 78 ∘C.

What is the final temperature of the water after the ice has melted? ∘C

To find the final temperature of the water after the ice has melted, we need to consider the heat gained by the water and the heat lost by the ice during the process.

First, let's calculate the heat gained by the water. We'll use the specific heat capacity of water, which is approximately 4.18 J/g⋅°C:

Q_water = m_water * c_water * ΔT_water

Where:

m_water = mass of water = 1.0 kg = 1000 g

c_water = specific heat capacity of water = 4.18 J/g⋅°C

ΔT_water = change in temperature of water = final temperature - initial temperature

Since we want to find the final temperature of the water, we'll assume it's the same as the final equilibrium temperature.

Now, let's calculate the heat lost by the ice. The heat lost by the ice is equal to the heat gained by the water, but with the opposite sign:

Q_ice = -Q_water

The heat lost by the ice is used to melt the ice, and the heat required to melt 1 gram of ice is called the heat of fusion, which is approximately 334 J/g.

Now, we'll set up the equation:

Q_ice = m_ice * L_fusion

Where:

m_ice = mass of ice = 116 g

L_fusion = heat of fusion of ice = 334 J/g

Since the heat lost by the ice is equal to the heat gained by the water, we can equate the two equations:

m_ice * L_fusion = -m_water * c_water * ΔT_water

Now, let's solve for ΔT_water:

ΔT_water = (m_ice * L_fusion) / (-m_water * c_water)

Substituting the given values:

ΔT_water = (116 g * 334 J/g) / (-(1000 g) * (4.18 J/g⋅°C))

Simplifying:

ΔT_water ≈ -9.09 °C

The negative sign indicates that the final temperature of the water will be lower than its initial temperature.

To find the final temperature, we subtract the magnitude of ΔT_water from the initial temperature of the water:

Final temperature = Initial temperature- |ΔT_water|

Final temperature = 78 °C - |-9.09 °C|

Final temperature[tex]≈ 68.91 °C[/tex]

Therefore, the final temperature of the water, after the ice has melted, is approximately 68.91 °C.

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