Problem 95 - Chemistry The Molecular Science

Question

You add \(100.0 \mathrm{~g}\) water at \(60.0^{\circ} \mathrm{C}\) to \(100.0 \mathrm{~g}\) ice at \(0.00^{\circ} \mathrm{C}\). Some of the ice melts and cools the water to \(0.00^{\circ} \mathrm{C}\). Calculate what mass of ice has melted when the ice and water mixture reaches a uniform temperature $$ \text { of } 0^{\circ} \mathrm{C} $$

Step-by-Step Solution

Answer

Approximately 75.12 grams of ice melts.

Step 1: Understand the Heat Exchange

In this problem, we are dealing with a heat exchange between warm water and ice. The water will release heat until it cools to the freezing point of ice, and this heat will be used to melt the ice.

Step 2: Identify the Relevant Formulas

The specific heat capacity of water is required to calculate the amount of heat lost by the water. The latent heat of fusion is required to calculate how much ice melts. We will use the formula for heat transfer: \[ Q = mc\Delta T \] And the formula for latent heat:\[ Q = mL \] where \( Q \) is the heat, \( m \) is the mass, \( c \) is the specific heat capacity, \( \Delta T \) is the change in temperature, and \( L \) is the latent heat of fusion of ice.

Step 3: Calculate Heat Released by Water

We'll first calculate the heat lost by the water as it cools from \(60.0^{\circ} \mathrm{C}\) to \(0.0^{\circ} \mathrm{C}\) using:\[ Q_w = m_w c_w \Delta T_w \]Here, \( m_w = 100.0 \text{ g} = 0.100 \text{ kg} \), \( c_w = 4.18 \text{ J/g°C} \), and \( \Delta T_w = 60.0^{\circ} \text{C} \). Therefore,\[ Q_w = 0.100 \times 4.18 \times 60 = 25.08 \text{ kJ} \]

Step 4: Calculate Ice Melted by Heat

The heat from the water will be used to melt the ice. To find the mass of the ice melted, we use:\[ Q_i = m_i L_f \]where \( Q_i = 25.08 \text{ kJ} \) and \( L_f = 334 \text{ J/g} \). Convert \( Q_i \) to J by multiplying by 1000 and solve for \( m_i \):\[ 25,080 = m_i \times 334 \]\[ m_i = \frac{25,080}{334} \approx 75.12 \text{ g} \]

Step 5: Conclusion

The mass of the ice that melts, causing the system to reach a uniform temperature of \(0.0^{\circ} \text{C}\), is approximately \(75.12 \text{ grams} \).

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy from a warmer object to a cooler one. In this scenario, warm water transfers heat to cold ice. This energy exchange is crucial in changing the state of substances. When heat is transferred:

• The warmer object (water) loses energy.
• The cooler object (ice) absorbs this energy.

This transfer is governed by straightforward calculations, making it a key concept in thermodynamics. Understanding heat transfer helps in predicting how systems reach equilibrium. When dealing with phase changes, such as melting, special formulas are needed to account for the energy used not just in heating but transforming states.

Specific Heat Capacity

Specific heat capacity is a property that describes how much energy is needed to change the temperature of a given mass of a substance. For water, this value is high: 4.18 J/g°C.

Why is specific heat capacity important?

• It determines how much energy is needed to change the water's temperature.
• In our problem, it helps calculate the energy lost by water as it cools from 60°C to 0°C.

Formula: \[ Q = mc\Delta T \]Where:

• \( Q \) is the energy in joules.
• \( m \) is the mass.
• \( c \) is the specific heat capacity.
• \( \Delta T \) is the change in temperature.

This relationship tells us how energy is distributed during temperature changes.

Latent Heat of Fusion

The latent heat of fusion is the energy required to change a substance from solid to liquid at its melting point. It's a unique property for each material. For ice, this value is 334 J/g.

Understanding latent heat helps explain why ice melts when absorbing heat:

• Even at 0°C, energy is needed to change ice to liquid water without increasing temperature.
• This energy comes from the warm water cooling down.

Formula: \[ Q = mL \]Where:

• \( Q \) is the heat absorbed.
• \( m \) is the mass of ice.
• \( L \) is the latent heat of fusion.

In the problem, this principle explains how much ice melts as the system equilibrates. It's essential for predicting how substances behave at melting or boiling points. Understanding this allows you to calculate changes in a system involving phase transitions.