Answer :

To determine how much of the ice melts when immersed in water, we need to examine the energy exchange between the ice and the water. The system will achieve thermal equilibrium, meaning the temperature will balance out.

Here are the steps to solve this problem:

  1. Identify Given Data:

    • Mass of ice, [tex]m_{ice} = 1.64[/tex] kg
    • Initial temperature of ice, [tex]T_{ice} = 0.00^\circ \text{C}[/tex]
    • Mass of water, [tex]m_{water} = 7.83[/tex] kg
    • Initial temperature of water, [tex]T_{water} = 9.51^\circ \text{C}[/tex]
    • Latent heat of fusion for ice, [tex]L_f = 334,000[/tex] J/kg
    • Specific heat capacity of water, [tex]c_{water} = 4,186[/tex] J/kg°C

  2. Calculate the energy required to melt the ice:

    The energy needed to melt the entire ice block is calculated using the formula:
    [tex]Q_{melt} = m_{ice} \times L_f[/tex]
    [tex]Q_{melt} = 1.64 \times 334,000[/tex]
    [tex]Q_{melt} = 547,760 \text{ J}[/tex]

  3. Calculate the energy released by cooling the water to 0°C:

    The water cools from 9.51°C to 0°C. The energy released is:
    [tex]Q_{cool} = m_{water} \times c_{water} \times \Delta T[/tex]
    [tex]\Delta T = 9.51° - 0° = 9.51°[/tex]
    [tex]Q_{cool} = 7.83 \times 4,186 \times 9.51[/tex]
    [tex]Q_{cool} = 311,452.5 \text{ J}[/tex]

  4. Determine how much ice melts:

    Since [tex]Q_{cool} < Q_{melt}[/tex], not all the ice will melt. The energy [tex]Q_{cool}[/tex] will only partially melt the ice. The mass of ice that melts can be calculated by dividing [tex]Q_{cool}[/tex] by the latent heat:
    [tex]m_{melted} = \frac{Q_{cool}}{L_f}[/tex]
    [tex]m_{melted} = \frac{311,452.5}{334,000}[/tex]
    [tex]m_{melted} \approx 0.932 \text{ kg}[/tex]

Therefore, approximately 0.932 kg of the ice melts.

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Rewritten by : Barada