Problem 40 - Chemistry The Central Science In Si Units

Question

The enthalpy of fusion of water is \(6.01 \mathrm{~kJ} / \mathrm{mol}\). Sunlight striking Earth's surface supplies \(168 \mathrm{~W}\) per square meter \((1 \mathrm{~W}=1 \mathrm{watt}=1 \mathrm{~J} / \mathrm{s}) .\) (a) Assuming that melting of ice is due only to energy input from the Sun, calculate how many grams of ice could be melted from a 1.00 square meter patch of ice over a \(12-\mathrm{h}\) day. \((\mathbf{b})\) The specific heat capacity of ice is \(2.032 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\). If the initial temperature of a 1.00 square meter patch of ice is \(-5.0^{\circ} \mathrm{C},\) what is its final temperature after being in sunlight for \(12 \mathrm{~h}\), assuming no phase changes and assuming that sunlight penetrates uniformly to a depth of \(1.00 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

(a) Approximately 21,750 g of ice could be melted. (b) The final temperature is approximately -1.0°C.

Step 1: Convert Sunlight Energy to Total Energy

First, find the total energy received by a 1.00 square meter area in 12 hours. The power per square meter is given as 168 W, which is equivalent to 168 J/s. Calculate the total energy over 12 hours: \( E = 168 \, \text{J/s} \times 12 \, \text{hours} \times 3600 \, \text{s/hour} = 7,257,600 \, \text{J} \).

Step 2: Calculate Moles of Ice Melted

Use the enthalpy of fusion to find how many moles of ice can be melted with the total energy calculated. The enthalpy of fusion is 6.01 kJ/mol, which converts to 6010 J/mol. Thus, \( \text{moles of ice} = \frac{7,257,600 \, \text{J}}{6010 \, \text{J/mol}} \approx 1208 \, \text{mol} \).

Step 3: Convert Moles to Grams

Convert the moles of ice to grams using the molar mass of water (18.015 g/mol): \( \text{grams of ice} = 1208 \, \text{mol} \times 18.015 \, \text{g/mol} \approx 21,750 \, \text{g} \).

Step 4: Calculate Total Mass of Ice

To find the final mass, considering the patch has a depth of 1.00 cm, calculate the total mass of ice in the patch using its density (approximately 917 g/L): \( \text{mass of ice}= 10000 \, \text{cm}^3 \times 0.917 \, \text{g/cm}^3 = 9170 \, \text{g} \).

Step 5: Calculate Temperature Change for Ice

Evaluate whether the entire patch of ice melts. If it doesn't, use the specific heat of ice \(c = 2.032 \, \text{J/g}°\text{C}\) to find the temperature change: \( \Delta T = \frac{7,257,600 \, \text{J}}{9170 \, \text{g} \times 2.032 \, \text{J/g}°\text{C}} \approx 4.00°\text{C} \).

Step 6: Determine Final Temperature

Calculate the final temperature of the ice, starting from -5.0°C: \( T_{\text{final}} = -5.0°C + 4.00°C = -1.0°C \).

Key Concepts

Melting of Ice

The melting of ice is an intriguing process where solid ice turns into liquid water. This transformation occurs when energy is added to the ice, usually in the form of heat. For ice to melt, it needs to reach its melting point, which for water is 0°C or 32°F.

During this process, energy from the environment is absorbed by the ice. This energy doesn't increase the temperature of the ice but is used to break the bonds holding the ice molecules in a solid structure. This concept is known as the enthalpy of fusion. In the presented problem, the enthalpy of fusion for ice is given as 6.01 kJ/mol. This means, to melt one mole of ice, 6.01 kJ of energy is required.

It's important to understand that the energy required to melt ice doesn't cause a temperature rise until all the ice has turned into water. Therefore, as sunlight strikes the icy surface, the absorbed energy can potentially melt significant amounts of ice, provided it's sufficient to overcome the enthalpy of fusion.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is needed to change the temperature of a certain mass of a substance by 1°C. Different substances require different amounts of energy for this temperature change.

For ice, the specific heat capacity is given as 2.032 J/g°C. This means that it takes 2.032 joules of energy to raise the temperature of one gram of ice by 1°C. This information helps us understand how ice responds to solar energy before it reaches its melting point. If the energy provided is not enough to melt the ice, it will be used to raise the temperature of the ice.

Understanding the specific heat capacity is important because, in any scenario where energy is being transferred to a substance, specific heat will determine how the substance's temperature changes with the amount of energy it absorbs. For example, sunlight in the exercise increases the temperature of ice until it can melt.

Energy Conversion

Energy conversion is a fundamental concept in understanding how ice melts under sunlight. Here, solar energy, provided by the Sun, converts into thermal energy when it strikes the ice. This thermal energy is then absorbed by the ice.

The rate of energy delivery from the sunlight is quantified in watts. For the exercise, it is specified as 168 W per square meter. A watt is a unit of power, equivalent to one joule per second. Thus, the energy received by the ice in a day can be calculated by multiplying this power by the total seconds in 12 hours (which is the duration the sunlight affects the ice).

Understanding energy conversion is crucial as it determines how efficiently the sunlight can melt the ice, or at least increase its temperature. Without this conversion, it would be difficult to predict the effects of solar energy on a patch of ice.

Temperature Change

Temperature change is the result of energy absorption by a substance and is a critical aspect when examining how solar energy affects ice.

When the icy surface absorbs the heat, if the energy isn't enough to melt the ice, the temperature of the ice will change. This change is calculated using the total energy supplied and the specific heat capacity of the ice. By dividing the total energy by the product of the mass of the ice and its specific heat capacity, the change in temperature can be determined.

In the exercise, the ice's temperature initially at -5.0°C increases due to sunlight. The final temperature is calculated considering no phase changes occur, meaning not all energy is used for melting. Instead, it raises the temperature. This illustrates how energy can influence both phase change and temperature, guiding us in understanding how much environment-driven energy affects natural elements like ice patches.