Question

Ice at \(0.0^{\circ} \mathrm{C}\) is mixed with \(5.00 \times 10^{2} \mathrm{mL}\) of water at \(25.0^{\circ} \mathrm{C} .\) How much ice must melt to lower the water temperature to \(0.0^{\circ} \mathrm{C} ?\)

Short answer

Approximately 156.41 grams of ice must melt to lower the water temperature to 0.0°C.

Step-by-step solution

Understand Heat Transfer

When ice at 0°C is added to water at a higher temperature, heat is transferred from the warmer water to the ice. The goal is to find how much ice must melt to lower the water temperature to 0°C.

Define Variables and Constants

The density of water is approximately 1 g/mL. The initial volume of water is 500 mL, which means its mass is 500 g. The specific heat of water, c, is 4.18 J/g°C. Latent heat of fusion for ice, L, is 334 J/g.

Calculate Heat Loss from Water

Since the initial temperature of water is 25°C and it is cooled to 0°C, the heat loss from water is calculated using the formula: \[ q_{water} = m imes c imes \Delta T \]Substitute: \[ q_{water} = 500 imes 4.18 imes (0 - 25) = -52,250 \, \text{J} \]

Relate Heat Gain by Ice to Heat Lost by Water

The heat gained by the ice will equal the heat lost by the water, according to the principle of conservation of energy. Thus, \[ q_{ice} = q_{water} \]

Calculate Mass of Ice that Melts

Using the formula for the heat absorbed by the melting ice:\[ q_{ice} = m_{ice} \times L \]We know \[ q_{ice} = 52,250 \, \text{J} \] and \[ L = 334 \, \text{J/g} \], so:\[ m_{ice} = \frac{52,250}{334} \approx 156.41 \, \text{g} \]

Key concepts

Latent Heat of Fusion

Latent heat of fusion is the heat absorbed or released by a substance during a change in its state, such as from a solid to a liquid, without changing its temperature. This occurs at the melting point of the substance. For ice, this is essential since we are considering how much ice must melt.
Understanding this concept is crucial for calculating how much heat energy is involved when ice melts. In this exercise, we are specifically using the latent heat of fusion for ice, which is 334 J/g.
When ice at 0°C is mixed with warmer water, the energy required for the ice to melt comes from the surrounding water. Since the process involves a change of state from solid to liquid, the energy used does not increase the temperature of the ice; it simply changes the state. To determine how much ice needs to melt, one must calculate the amount of energy required using the formula:

• The equation used is \[ q_{ice} = m_{ice} \times L \].
• Here, \( q_{ice} \) is the heat absorbed by the melting ice.
• \( m_{ice} \) is the mass of the ice.
• \( L \) is the latent heat of fusion.

To find the mass of ice that melts, you rearrange this formula and plug in the known values, allowing you to determine how much ice is necessary to absorb the excess heat from the warm water, thereby cooling it down to 0°C.

Specific Heat Capacity

Specific heat capacity is an important property of materials that measures the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. In simple words, it tells us how much heat energy needs to be added to a substance before it gets hotter.
This concept is vital when looking at how much heat is transferred in processes involving temperature changes. For water, the specific heat capacity is 4.18 J/g°C.
The specific heat capacity is used to calculate how much energy the water loses as it cools from 25°C to 0°C. This is done using the formula:

• The equation is \[ q_{water} = m \times c \times \Delta T \].
• Where \( m \) is the mass of the water.
• \( c \) is the specific heat capacity.
• \( \Delta T \) is the change in temperature.

This allows us to compute the total heat loss.By using this information, you can understand how much heat energy needs to be transferred to or from the substance to change its temperature. It gives insight into the efficiency of the cooling or heating process, particularly when related to large quantities of water, as seen in this textbook problem.

Conservation of Energy

The principle of conservation of energy holds that energy cannot be created or destroyed but only transferred from one form to another or from one body to another. In thermal processes, the total amount of heat energy before and after the process remains constant.
This principle helps us understand the relationship between the water and the ice in this exercise. When the warm water cools down, the energy it loses goes into melting the ice.
Thus, the heat energy lost by the water must be equal to the heat gained by the ice melting. This equality is the core assumption which connects heat loss from the water \( q_{water} \) to the heat gain by melting ice \( q_{ice} \) with the equation:

• The expression representing this is \[ q_{ice} = q_{water} \].

In practical terms, when calculating how much ice melts, the energy conservation idea ensures that every joule lost by the water is accounted for by ice gaining heat and melting, maintaining a balanced energy exchange.