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Correct Answer: 0 oc, 2.0 g ice, 18.0g water.
to solve this problem, we need to calculate the energy transfers involved when mixing ice and water at different temperatures, ensuring energy conservation principles are maintained (i.e., the total energy before mixing equals the total energy after mixing).
we start with 8 grams of ice at 0°c and 12 grams of water at 40°c. let's first find out how much energy would be released by the water as it cools down to 0°c, and then check whether this energy is sufficient to melt all the ice.
**step 1: calculate the energy released by the cooling water**
the specific heat capacity of water is 4.18 j/g°c. the formula for calculating the energy change due to a temperature change is:
\[
q = m \times c \times \delta t
\]
where:
- \( q \) is the heat absorbed or released,
- \( m \) is the mass of the substance,
- \( c \) is the specific heat capacity,
- \( \delta t \) is the change in temperature.
for the cooling water:
\[
q_{water} = 12 \text{ g} \times 4.18 \text{ j/g°c} \times (40°c - 0°c) = 2006.4 \text{ j}
\]
this is the energy released by the water as it cools to 0°c.
**step 2: calculate the amount of ice that can be melted by this energy**
the latent heat of fusion of ice is 334 j/g, which is the energy required to melt one gram of ice into water at the same temperature (0°c). the amount of ice that can be melted by the energy released from the cooling water can be calculated by:
\[
m_{ice\_melted} = \frac{q_{water}}{\text{latent heat of fusion}} = \frac{2006.4 \text{ j}}{334 \text{ j/g}} \approx 6.007 \text{ g}
\]
rounding this off gives approximately 6.0 g of ice that can be melted.
**step 3: determine the final state of the system**
initially, we have 8 g of ice. if 6.0 g of it melts due to the energy released by the water, we're left with:
\[
8 \text{ g} - 6.0 \text{ g} = 2.0 \text{ g}
\]
so, 2.0 g of ice remains un-melted.
the melted ice (6.0 g) plus the original water (12 g) results in:
\[
6.0 \text{ g} + 12 \text{ g} = 18.0 \text{ g}
\]
thus, the final mixture contains 2.0 g of ice and 18.0 g of water, all at 0°c. this conclusion ensures that the principle of conservation of energy is satisfied, as all the heat lost by the water has been used to melt part of the ice, and the final temperature of the mixture stabilizes at the melting point of ice, which is 0°c.
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