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 \((1 \mathrm{W}=1\) 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 -h day. (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 \(\mathrm{cm} ?\)

Short answer

(a) The mass of ice that can be melted in a 1.00 square meter patch of ice over a 12-hour day is 21744.77 grams. (b) The final temperature we calculated is 387.24°C, but this is not possible in real-life, as the ice patch would melt before reaching this temperature. The energy would instead contribute to the phase change or heating the surrounding environment.

Step-by-step solution

(a) Calculate the energy provided by sunlight over 12 hours

First, we should find the total energy supplied by sunlight per square meter over a 12-hour day. We know that sunlight provides 168 W per square meter, and we can use the formula: Energy (J) = Power (W) × Time (s) The time duration is 12 hours, so we need to convert it into seconds: 12 hours × (3600 seconds/hour) = 43200 seconds Now, we can calculate the energy provided by sunlight: Energy = 168 W × 43200 s = 7257600 J

(a) Calculate the mass of ice that can be melted

Next, we will use the enthalpy of fusion to determine how many grams of ice can be melted. The enthalpy of fusion of water is given as 6.01 kJ/mol, and we should convert it to J/mol: 6.01 kJ/mol × (1000 J/kJ) = 6010 J/mol To find the mass of ice, we need to divide the energy provided by sunlight by the enthalpy of fusion per mole of ice. First, we find the number of moles of ice that can be melted: Moles of ice = Energy / (Enthalpy of fusion per mole) = 7257600 J / 6010 J/mol = 1207.82 mol Now, we can find the mass of ice using the molar mass of water (18.015 g/mol): Mass of ice = Moles of ice × Molar mass of water = 1207.82 mol × 18.015 g/mol = 21744.77 g So, the mass of ice that can be melted in a 1.00 square meter patch of ice over a 12-hour day is 21744.77 grams.

(b) Calculate the energy absorbed by the ice patch

In this part, we will assume no phase change and calculate the energy absorbed by the ice patch. We are given the specific heat capacity of ice (2.032 J/g°C) and the depth of sunlight penetration (1.00 cm). Let's first find the volume of the ice patch: Volume = Area × Depth = 1.00 m² × 0.01 m = 0.01 m³ Next, we need to convert it to grams, using the density of ice (0.917 g/cm³): Mass of ice = Volume × Density We need to convert the volume to cm³: 0.01 m³ × (1,000,000 cm³/m³) = 10,000 cm³ Mass of ice = 10,000 cm³ × 0.917 g/cm³ = 9170 g

(b) Calculate the final temperature of the ice patch

We can now use the energy provided by sunlight, the mass of ice, and the specific heat capacity of ice to calculate the change in temperature: Change in temperature = Energy / (Mass × Specific heat capacity) = 7257600 J / (9170 g × 2.032 J/g°C) = 392.24 °C Since the initial temperature of the ice patch is -5.0°C, the final temperature after 12 hours can be found by adding the change in temperature to the initial temperature: Final temperature = Initial temperature + Change in temperature = -5.0°C + 392.24°C = 387.24°C However, ice melts at around 0°C, so the final temperature we calculated is not possible in real-life. The ice patch will instead melt before reaching that temperature, and any remaining energy will contribute to the phase change or heating the surrounding environment.

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity is a measure of how much energy is required to raise the temperature of one gram of a substance by one degree Celsius. It's an intrinsic property, meaning it is a characteristic of the substance itself, not the amount of the substance.

For example, the specific heat capacity of ice is 2.032 J/g°C, which means it takes 2.032 joules of energy to increase the temperature of one gram of ice by one degree Celsius. This value is crucial in calculating the amount of energy needed to change the temperature of any mass of ice. In our exercise, we leveraged this concept to determine the energy required to raise the temperature of a given mass of ice before reaching its melting point.

Phase Change and Energy

A phase change is the transformation of a substance from one state of matter to another, such as from solid to liquid or liquid to gas. Along with temperature changes, phase changes require energy, but this energy doesn't lead to a temperature increase; it's used to overcome intermolecular forces that hold the particles together in a particular phase.

During the melting process, or fusion, ice requires a certain amount of heat energy, known as the enthalpy of fusion, to change from solid to liquid at its melting point without changing temperature. The enthalpy of fusion for water is 6.01 kJ/mol, signifying the amount of energy needed to melt one mole of ice. This is the energy that must be provided by the sun's rays to melt ice in the exercise we're examining.

Energy Calculations in Chemistry

In chemistry, the calculation of energy involved in temperature changes and phase changes is essential for understanding heat transfer during these processes. Using the enthalpy of fusion and specific heat capacity, we can estimate how much energy is needed to achieve a particular temperature change or to melt a certain mass of a substance.

Converting energy units and using correct formulas help to precisely determine the effects of energy input from an external source, like the sun in our exercise. The calculations provided not solely allow us to predict the outcome but also offer insights into energy conservation and exchange within a system, highlighting the importance of understanding energy calculations to interpret various physical and chemical phenomena in nature.