This problem involves the fundamental principle of heat transfer, where thermal energy is exchanged between objects at different temperatures. Specifically, the heat lost by the hot copper block is absorbed by the ice, causing it to melt.

Copper Block Heat Transfer to Ice

When a hot copper block is placed on an ice block, the copper block will release its thermal energy, cooling down until it reaches the melting point of ice, which is 0°C. This released heat energy is then absorbed by the ice, causing a portion of it to change from a solid state to a liquid state (melt) without a change in temperature (as long as both ice and water coexist at 0°C).

Understanding Heat Transfer Principles

• Heat Lost by Copper: The amount of heat lost by the copper block depends on its mass, specific heat capacity, and the change in its temperature. This is calculated using the formula: $$Q_{\text{lost}} = m_{\text{copper}} \times c_{\text{copper}} \times \Delta T_{\text{copper}}$$ where \(m_{\text{copper}}\) is the mass of the copper, \(c_{\text{copper}}\) is its specific heat, and \(\Delta T_{\text{copper}}\) is the temperature change.
• Heat Gained by Ice: The amount of heat required to melt a certain mass of ice at its melting point is determined by its mass and the latent heat of fusion. This is calculated using the formula: $$Q_{\text{gained}} = m_{\text{ice}} \times L_f$$ where \(m_{\text{ice}}\) is the mass of ice melted and \(L_f\) is the latent heat of fusion of water.
• Conservation of Energy: Assuming no heat is lost to the surroundings, the heat lost by the copper block is entirely absorbed by the ice. Therefore: $$Q_{\text{lost}} = Q_{\text{gained}}$$

Given Parameters for Copper Block

• Mass of copper block (\(m_{\text{copper}}\)): 3 kg = 3000 g
• Initial temperature of copper block (\(T_{\text{initial, copper}}\)): 450°C
• Final temperature of copper block (\(T_{\text{final, copper}}\)): 0°C (because it cools down to the melting point of ice)
• Change in temperature (\(\Delta T_{\text{copper}}\)): \(T_{\text{initial, copper}} - T_{\text{final, copper}} = 450\text{°C} - 0\text{°C} = 450\text{°C}\). Since a change in Celsius is equivalent to a change in Kelvin, \(\Delta T_{\text{copper}} = 450\text{ K}\).
• Specific heat of copper (\(c_{\text{copper}}\)): 0.39 Jg\(^{-1}\)K\(^{-1}\)

Given Parameters for Ice Block

• Heat of fusion of water (\(L_f\)): 335 Jg\(^{-1}\)
• Initial temperature of ice: 0°C (as it is an ice block at its melting point)

Calculating Heat Lost by Copper

First, we need to calculate the total amount of heat energy released by the copper block as it cools from 450°C to 0°C.

Using the formula: $$Q_{\text{lost}} = m_{\text{copper}} \times c_{\text{copper}} \times \Delta T_{\text{copper}}$$

Substitute the given values into the formula:

$$Q_{\text{lost}} = 3000\text{ g} \times 0.39\text{ Jg}^{-1}\text{K}^{-1} \times 450\text{ K}$$

$$Q_{\text{lost}} = 1170\text{ J K}^{-1} \times 450\text{ K}$$

$$Q_{\text{lost}} = 526500\text{ J}$$

Calculating Mass of Ice Melted

Now, this heat energy lost by the copper block is completely absorbed by the ice to melt it. We can use the latent heat of fusion formula to find the mass of ice melted.

The formula for heat absorbed by ice to melt is: $$Q_{\text{gained}} = m_{\text{ice}} \times L_f$$

Since \(Q_{\text{lost}} = Q_{\text{gained}}\), we can set up the equation:

$$526500\text{ J} = m_{\text{ice}} \times 335\text{ Jg}^{-1}$$

To find the mass of ice melted (\(m_{\text{ice}}\)), rearrange the equation:

$$m_{\text{ice}} = \frac{526500\text{ J}}{335\text{ Jg}^{-1}}$$

$$m_{\text{ice}} = 1571.64179\dots \text{ g}$$

To express the mass in kilograms, divide by 1000:

$$m_{\text{ice}} = \frac{1571.64179\dots \text{ g}}{1000\text{ g/kg}}$$

$$m_{\text{ice}} \approx 1.5716\text{ kg}$$

Final Result

The maximum amount of ice that can melt due to the heat transferred from the copper block is approximately 1.57 kg.