Problem 29 - General Chemistry: Principles and Modern Applications

Question

What mass of ice can be melted with the same quantity of heat as required to raise the temperature of \(3.50 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}(1)\) by \(50.0^{\circ} \mathrm{C} ?\left[\Delta H_{\text {fusion }}^{\circ}=6.01 \mathrm{kJ} / \mathrm{mol}\right.\) \(\left.\mathrm{H}_{2} \mathrm{O}(\mathrm{s})\right]\)

Step-by-Step Solution

Answer

The number of moles of ice that could be melted using the same amount of heat required to raise the temperature of \(3.50 \mathrm{mol}\) water by \(50.0^{\circ} \mathrm{C}\) can be calculated using the heat of fusion and standard specific heat principles. Then, convert that result from moles to grams using the molar mass of water

Step 1: Calculate the heat required to raise the temperature of water

Firstly, let's find out how much heat is needed to raise the temperature of liquid water by \(50.0^{\circ} \mathrm{C}\). This can be calculated using the formula Q=mcΔT. Where Q represents heat energy, m is the number of moles, c refers to the specific heat capacity of water (4.18 kJ/mol•K), and ΔT is the change in temperature (50°C in this case). Thus, Q needed for water: \(Q = 3.50 \mathrm{mol} * 4.18 \mathrm{kJ/mol•K} * 50 \mathrm{K}\)

Step 2: Calculate the heat required to melt ice

Next, let's calculate how much heat is needed to change the state of water from solid (ice) to liquid using the same heat found in step 1. This can be done using the formula Q=ΔHfusion*n, where ΔHfusion is the heat of fusion of water (6.01 kJ/mol) and n is the number of moles. Thus, to find out the mass of ice that can be melted, we need to rearrange the formula to n = Q / ΔHfusion, and plug in the values.

Step 3: Adjust for grams

Lastly, remember that to convert the number of moles to grams, we need to multiply number of moles by the molar mass of water (18 g/mol). Thus, the mass of ice that could be melt is: Mass = n * molar mass of water

Key Concepts

Heat transfer

Heat transfer refers to the movement of thermal energy from one object or substance to another. In our exercise, heat energy is being transferred to increase the temperature of liquid water. This occurs through a process called conduction, where heat passes through a material without moving the material itself. The formula used for calculating this transfer is:

• \( Q = mc\Delta T \)

Here, \( Q \) represents the heat energy transferred, \( m \) is the number of moles of the substance, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. In this exercise, we need to raise the temperature of 3.50 moles of water by 50°C. This requires a specific amount of energy, which we calculate based on the formula provided. This process is crucial in understanding how energy is absorbed or released during chemical changes in everyday phenomena. Learning about heat transfer is essential in fields such as engineering, meteorology, and even cooking. Understanding how to calculate this transfer is fundamental in thermodynamics.

Phase change

Phase change occurs when a substance changes from one state of matter to another, such as from solid to liquid or from liquid to gas. In this problem, we focused on the phase transition from solid ice to liquid water. This transformation happens when a specific amount of heat is absorbed by the ice without changing its temperature. The latent heat involved in this process is known as the heat of fusion. During phase changes, energy is absorbed or released, but there's no change in temperature until the phase transition is complete. The formula used for determining the heat required for a phase change is:

• \( Q = \Delta H_{\text{fusion}} \times n \)

Where \( Q \) is the heat absorbed, \( \Delta H_{\text{fusion}} \) is the heat of fusion, and \( n \) is the number of moles. In our exercise, we calculate how much mass of ice can be melted with the heat energy that was calculated in the heat transfer section. Phase changes are vital as they involve energy efficiency and conservation principles, which are applied in various fields, including climate science and refrigeration.

Calorimetry

Calorimetry is the science of measuring the amount of heat involved in chemical reactions or physical changes. It's a fundamental technique used in thermodynamics to understand and quantify energy transfer processes. In this exercise, calorimetry allows us to calculate the heat needed to raise the temperature of water and to melt ice, helping us determine the key components involved in both processes. The basic principle behind calorimetry is the use of a calorimeter—a device designed to minimize energy exchange with the external environment. This ensures accurate measurements of heat transfer and energy changes within the system being studied. Common steps in calorimetry include:

• Selecting the appropriate reaction or process.
• Measure initial conditions (like temperature).
• Allow the reaction or phase change to occur and measure final conditions.
• Calculate the heat exchange using calorimetric principles.

In everyday applications, calorimetry helps in understanding dietary energy, fuel combustion, and industrial processes. Recognizing how to measure and analyze heat flow can be beneficial in diverse fields from food science to engineering.