Question

The enthalpy of evaporation of water is \(40.67 \mathrm{~kJ} / \mathrm{mol}\). 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 evaporation of water is only due to energy input from the Sun, calculate how many grams of water could be evaporated from a 1.00 square meter patch of ocean over a 12 -hour day. (b) The specific heat capacity of liquid water is \(4.184 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\). If the initial temperature of a 1.00 square meter patch of ocean is \(26^{\circ} \mathrm{C},\) what is its final temperature after being in sunlight for 12 hours, assuming no phase changes and assuming that sunlight penetrates uniformly to depth of \(10.0 \mathrm{~cm}\) ?

Short answer

In summary, (a) 3,219.59 grams of water could be evaporated from a 1.00 square meter patch of ocean over a 12-hour day, and (b) the final temperature of the 1.00 square meter patch of ocean after being in sunlight for 12 hours and assuming no phase changes is 43.3°C.

Step-by-step solution

Part (a): Calculate the energy received from sunlight

To calculate the energy received from sunlight, we'll use the formula: Energy = Power × Time We're given that the sunlight supplies 168 W per square meter and we're looking at a 1 square meter area for 12 hours. Time = 12 hours × 3600 seconds/hour = 43,200 seconds So the Energy from sunlight in 12 hours is: Energy = (168 J/s) × 43,200 s = 7,257,600 Joules

Part (a): Calculate the mass of water evaporated

To determine the mass of water evaporated, we will use the energy calculated above and the enthalpy of evaporation formula: Enthalpy of evaporation = (Energy / Mass of water) × Molar mass of water Given, the enthalpy of evaporation of water is 40.67 kJ/mol and molar mass of water is 18.015 g/mol. Rearrange the formula to calculate the mass of water, Mass of water = (Energy × Molar mass of water) / Enthalpy of evaporation Mass of water = (7,257,600 J × 18.015 g/mol) / (40.67 kJ/mol × 1000 J / kJ) Mass of water = 3,219.59 g Hence, 3,219.59 grams of water could be evaporated from a 1.00 square meter patch of ocean over a 12-hour day.

Part (b): Calculate the initial energy of the ocean patch

The initial energy can be calculated using the formula: Initial energy = Mass of water × Specific heat capacity × Initial temperature We are given that the initial temperature is 26°C and the specific heat capacity of water is 4.184 J/g°C. To calculate the mass of water in the patch, we need to use the volume and density of water: Density of liquid water = 1.0 g/cm³ Depth = 10.0 cm Area = 1.00 m² = 10000 cm² Volume of water in the patch = Area × Depth = 10000 cm² × 10.0 cm = 100,000 cm³ Mass of water in the patch = Volume × Density = 100,000 cm³ × 1.0 g/cm³ = 100,000 g Now, we can calculate the initial energy of the ocean patch: Initial energy = (100,000 g) × (4.184 J/g°C) × (26°C) = 10,879,200 J

Part (b): Calculate the energy of the ocean patch after 12 hours

The energy of the ocean patch after 12 hours of sunlight can be calculated by adding the energy supplied by sunlight: Energy after 12 hours = Initial energy + Energy from sunlight = 10,879,200 J + 7,257,600 J = 18,136,800 J

Part (b): Calculate the final temperature of the ocean patch

To calculate the final temperature of the ocean patch, we use the formula: Final temperature = Energy after 12 hours / (Mass of water × Specific heat capacity) Final temperature = (18,136,800 J) / (100,000 g × 4.184 J/g°C) Final temperature = 43.3°C Hence, the final temperature of the 1.00 square meter patch of ocean after being in sunlight for 12 hours and assuming no phase changes is 43.3°C.

Key concepts

Enthalpy of Evaporation

When water changes from a liquid to a gas, it needs energy. This energy is called the enthalpy of evaporation. It is the amount of energy needed to evaporate one mole of a substance. For water, this value is 40.67 kJ/mol. Knowing this helps us predict how much energy is necessary to evaporate a specific amount of water.

To calculate the mass of water that can evaporate with a given energy, we rearrange the formula:\[\text{Mass of water} = \frac{\text{Energy} \times \text{Molar mass of water}}{\text{Enthalpy of evaporation}}\]Where:

• Energy is from the Sun or another source.
• Molar mass of water: 18.015 g/mol

Using this formula, we can calculate how many grams of water can evaporate with a specific energy input from the Sun.

Specific Heat Capacity

Specific heat capacity is a vital concept in thermochemistry. It tells us how much energy is required to raise the temperature of a certain mass of a substance by one degree Celsius. For water, this value is 4.184 J/g°C, which means water requires 4.184 Joules per gram to increase its temperature by 1°C.

This property is why water heats and cools slowly compared to other substances. Its high specific heat capacity makes water an excellent thermal regulator. We use this concept to calculate how much the temperature of a body of water changes when energy is added, applying the formula: \[\text{Energy} = \text{Mass} \times \text{Specific heat capacity} \times \Delta T\]Where:

• \( \Delta T \) is the change in temperature.

Applying this equation allows us to calculate temperature changes in water bodies, given a known energy input.

Energy Calculation

Energy calculations are crucial to determine how much energy we are dealing with in various processes. Whether from the Sun or other sources, calculating energy helps us understand how systems change. In this context, we often calculate energy using power and time relationships. Power (watts) is converted to energy (Joules) over a period.

Simple formula for energy is:\[\text{Energy} = \text{Power} \times \text{Time}\]In our calculations, time is given in seconds because 1 watt = 1 joule/second. This transformation is helpful when understanding daily sunlight energy input or other scenarios. Knowing this relationship helps in estimating effects like temperature changes or evaporation rates in practical situations.

Solar Energy

Solar energy is the energy we receive from the Sun, and it plays a crucial role in various processes on Earth, such as climate systems and water cycles. It is abundant and clean, making it a focal point of renewable energy sources. When solar energy reaches Earth, it is measured as power per square meter, and typical sunlight provides about 168 watts per square meter on Earth’s surface.

This energy contributes significantly to evaporation processes, contributing to the water cycle by evaporating water in oceans, lakes, and rivers. By harnessing solar energy, we can estimate how much water can evaporate from a specific area or how it alters temperatures in a given ecosystem. Understanding solar energy metrics allows for better resource management and sustainable practices across scientific and engineering fields.