Chapter 17: Problem 58

Describe the composition and temperature of the equilibrium mixture after \(1.0 \mathrm{kg}\) of ice at \(-40^{\circ} \mathrm{C}\) is added to \(1.0 \mathrm{kg}\) of water at \(5.0^{\circ} \mathrm{C}\)

Short Answer

Expert verified

The final temperature and composition of the mixture depends on whether all the ice melts. If it does, the mixture will have a certain temperature above 0°C. Otherwise, the mixture will be at 0°C with some ice remaining. The actual values depend on the specific heat capacities and the heat of fusion of ice.

Step by step solution

Calculate the heat needed to warm the ice to 0°C

First, calculate the heat needed to warm the ice from -40°C to 0°C using the specific heat capacity of ice (\(2.09 \, J/(g \cdot °C)\)) and the formula \(Q=m \cdot c \cdot \Delta T\), where m is the mass (1000 g), c is the specific heat, and \(\Delta T\) is the temperature change (40°C).

Calculate the heat required to melt the ice

Next, calculate the heat required to melt the ice into water at 0°C using the formula \(Q=m \cdot L\), where m is the mass (1000 g), and L is the heat of fusion of ice (\(334 \, J/g\)).

Calculate the heat lost by the water

Now, calculate the heat lost by the water as it cools from 5°C to 0°C. This is calculated similarly to Step 1, but using the specific heat capacity of water (\(4.18 \, J/(g \cdot °C)\)).

Check if ice will melt completely

Add up the heat absorbed by the ice in Steps 1 and 2, and compare it to the heat lost by the water in Step 3. If the heat absorbed by the ice is less than or equal to the heat lost by the water, then all the ice will melt and the resulting mixture will be at 0°C. Otherwise, some ice will remain and the mixture will still be at 0°C, since the excess heat from the water will be used to melt as much ice as possible.

Calculate the final temperature if all ice melts

If all the ice melts, calculate the final temperature of the water-ice mixture by equating the heat gained by the melted ice (which is now water at 0°C) to the heat lost by the water, and solve for the final temperature. Use the formula \(Q=m \cdot c \cdot \Delta T\) and the specific heat capacity of water.

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Describe the composition and temperature of the equilibrium mixture after \(1.0 \mathrm{kg}\) of ice at \(-40^{\circ} \mathrm{C}\) is added to \(1.0 \mathrm{kg}\) of water at \(5.0^{\circ} \mathrm{C}\)

Short Answer

Step by step solution

Calculate the heat needed to warm the ice to 0°C

Calculate the heat required to melt the ice

Calculate the heat lost by the water

Check if ice will melt completely

Calculate the final temperature if all ice melts

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