Problem 54 - University Physics with Modern Physics

Question

A 4.00-kg silver ingot is taken from a furnace, where its temperature is 750.0\(^\circ\)C, and placed on a large block of ice at 0.0\(^\circ\)C. Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

Step-by-Step Solution

Answer

Approximately 2111 grams of ice is melted.

Step 1: Understand the Energy Transfer

The heat lost by the silver ingot will be absorbed by the ice, causing the ice to melt. We use the formula for heat transfer: \[ q = m \cdot c \cdot \Delta T \] where \( q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.

Step 2: Calculate the Heat Lost by Silver

The specific heat capacity of silver is approximately 0.235 J/g°C. The temperature change \( \Delta T \) for silver is from 750.0°C to 0.0°C, which is \( 750.0 - 0.0 = 750.0 \)°C.Convert the mass of silver from kilograms to grams: 4.00 kg = 4000 g.Now, substitute into the formula: \[ q = 4000 \cdot 0.235 \cdot 750.0 \]Calculate \( q \).

Step 3: Solve for Heat Energy

Now, let's perform the calculations:\[ q = 4000 \cdot 0.235 \cdot 750.0 = 705,000 \text{ J} \]So, the heat lost by the silver is 705,000 joules.

Step 4: Calculate Mass of Ice Melted

To find out how much ice is melted, use the latent heat of fusion for ice which is 334 J/g. The formula is:\[ m_{\text{ice}} = \frac{q}{L_f} \]where \( L_f \) is the latent heat of fusion. Substituting the values:\[ m_{\text{ice}} = \frac{705000}{334} \]Calculate \( m_{\text{ice}} \).

Step 5: Solve for Mass of Ice

Complete the calculation:\[ m_{\text{ice}} = \frac{705000}{334} \approx 2111 \text{ g} \]So, approximately 2111 grams of ice is melted.

Key Concepts

Specific Heat Capacity

Every substance has a specific heat capacity, which is essentially the amount of heat required to raise the temperature of 1 gram of the substance by 1°C. For silver, this specific heat capacity is approximately 0.235 J/g°C. This means that it takes 0.235 joules of energy to increase the temperature of 1 gram of silver by one degree Celsius. When an object, like the silver ingot from our exercise, is placed in a new environment, its heat capacity determines how much energy it can transfer to or absorb from that environment.
This concept is central to calculating how much heat an object can release. The formula used for calculating the heat ( q ) in this scenario is:

• q = m \cdot c \cdot \Delta T

where \(m\) is mass, \(c\) is specific heat capacity, and \(\Delta T\) is the change in temperature. Understanding how specific heat capacity works is crucial for solving problems involving temperature changes and energy transfer.

Latent Heat of Fusion

While specific heat capacity measures energy needed to alter temperature, latent heat of fusion describes the energy required to change a substance's state from solid to liquid without altering its temperature. For ice, this value is 334 J/g. This is the amount of energy needed to melt 1 gram of ice at its melting point, 0°C, into water at 0°C.
In our exercise, the energy released by the silver ingot as it cools is used to melt the ice. To calculate how much ice melts, we use the formula:

• m_{\text{ice}} = \frac{q}{L_f}

where \(q\) is the heat energy absorbed by the ice and \(L_f\) is the latent heat of fusion. This formula helps us determine that approximately 2111 grams of ice was melted by the ingot.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In physical systems, energy is constantly being transformed; for example, in our exercise, thermal energy from the silver is converted into the energy required to melt ice. This is an example of heat energy being utilized in a thermodynamic process.
Understanding energy conversion is crucial in many scientific and engineering calculations. It highlights the idea that energy is conserved and can only change forms, not be created or destroyed. When the hot silver ingot cools down, its stored thermal energy decreases and becomes useful energy to change the phase of the ice from solid to liquid.

Physical Chemistry

This branch of chemistry explores how matter behaves on a molecular and atomic level and explains the physical properties of matter. It uses principles of physics and chemistry to study how systems and processes behave.
The concepts in this exercise, like heat transfer and phase changes, are deeply rooted in physical chemistry. When you heat or cool a substance, you're observing physical reactions that are driven by these chemical physics concepts. Understanding how heat affects a substance, such as triggering a phase change or causing it to absorb or release energy, is fundamental to this field. The exercise showcases these principles by displaying how heat from the silver ingot leads to the melting of ice, which involves understanding not just the chemical makeup, but the physical properties too.

Thermodynamics

Thermodynamics is the study of heat, energy, and work. It examines how energy is transferred in physical processes and how it affects matter. This science is crucial for understanding why and how heat transfer occurs, such as in the example with our silver ingot and ice.
The first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed, is evident here. The system, in this case the silver ingot and the ice, must obey this principle. The energy from the silver does not disappear; instead, it's transferred to the ice to facilitate melting. Understanding these laws of energy distribution and conservation are vital in predicting the outcomes of thermal exchanges. Thermodynamics helps us comprehend not just this exercise, but broader phenomena in the natural world.