Problem 88 - Physics

Question

Two grams of liquid water are at \(0{ }^{\circ} \mathrm{C}\), and another two grams are at \(100{ }^{\circ} \mathrm{C}\). Heat is removed from the water at \(0{ }^{\circ} \mathrm{C}\), completely freezing it at \(0{ }^{\circ} \mathrm{C}\). This heat is then used to vaporize some of the water at \(100{ }^{\circ} \mathrm{C}\). What is the mass (in grams) of the liquid water that remains?

Step-by-Step Solution

Answer

1.70 grams of water remains as liquid.

Step 1: Calculate Heat Removed from Freezing Water

Two grams of water at 0°C are to be frozen. The heat removed, Q, can be calculated using the equation: \( Q = m \cdot L_f \), where \( L_f \) (latent heat of fusion for water) is 334 J/g. Therefore, \( Q = 2 \times 334 = 668 \) J.

Step 2: Calculate Mass of Water Vaporized

The heat removed is used to vaporize water at 100°C. Use the equation: \( m = \frac{Q}{L_v} \), where \( L_v \) (latent heat of vaporization for water) is 2260 J/g. Thus, \( m = \frac{668}{2260} \approx 0.29558 \) grams of water are vaporized.

Step 3: Calculate Remaining Liquid Water

Subtract the vaporized water mass from the original two grams at 100°C. \( 2 - 0.29558 = 1.70442 \) grams of liquid water remain.

Key Concepts

Latent Heat of Fusion

The latent heat of fusion is the amount of heat required to convert a solid into a liquid at the same temperature. For water, this is especially important because it's involved in the process of ice (solid water) turning into liquid water, and vice versa. The key point is that during the phase change from liquid to solid or solid to liquid, the temperature of the substance does not change even though heat is being absorbed or released.

For water, the latent heat of fusion is 334 J/g. This means that to turn 1 gram of ice at its melting point into liquid water at the same temperature, 334 joules of energy must be added. Conversely, to freeze 1 gram of liquid water at 0°C, 334 joules of energy must be removed.

In the exercise, 668 joules of energy are removed from 2 grams of water at 0°C to freeze it completely. This is calculated by multiplying the latent heat of fusion (334 J/g) by the mass of the water (2 grams): \( Q = m \cdot L_f = 2 \times 334 = 668 \) J.

Latent Heat of Vaporization

Latent heat of vaporization refers to the heat required to convert a liquid into a vapor at the same temperature. It is a critical concept in processes where a substance changes from liquid to gas without a change in temperature. This heat breaks the interactions between molecules in the liquid, allowing them to move freely as a gas.

For water, the latent heat of vaporization is 2260 J/g. This means to transform 1 gram of water into steam at its boiling point, 2260 joules of energy must be supplied. Conversely, to condense steam into water, the same amount of energy is released.

In the given problem, 668 joules of energy removed from the freezing process is used to vaporize water at 100°C. By using the formula \( m = \frac{Q}{L_v} \), where \( L_v \) is the latent heat of vaporization, we can determine how much water can be converted to steam: \( m = \frac{668}{2260} \approx 0.29558 \) grams are vaporized.

Phase Changes

Phase changes occur when a substance transforms from one state of matter to another, such as from solid to liquid or liquid to gas. During these changes, the temperature of the substance remains constant, as the heat energy is used to alter the state rather than increase temperature.

Two primary phase changes discussed here are

• Fusion: Solid to liquid (or liquid to solid), which involves the latent heat of fusion.
• Vaporization: Liquid to gas (or gas to liquid), involving the latent heat of vaporization.

It's important to remember that during any phase change, the energy added or removed changes the molecular organization, not the temperature. In the exercise, freezing and vaporization are the two phase changes, closely dictated by their respective latent heats.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. This process is crucial in evenly distributing heat within a system or transferring it from one phase of matter to another. There are three key methods of heat transfer:

• Conduction: Direct contact allows heat to flow between molecules, as seen in solids.
• Convection: Heat moves through liquids and gases caused by the movement of currents within the fluid.
• Radiation: Heat is transferred through electromagnetic waves, typical in sunlight warming a surface.

In the example exercise, heat is transferred from the liquid water at 0°C (as it freezes) to vaporize some of the water at 100°C. The amount of heat transferred allows us to quantitatively understand how much water is vaporized using the latent heat values. This interplay of heat transfer illustrates energy conservation within the closed system of the exercise.