Evaporating sweat cools the body because evaporation is an endothermic process: H2O(l) → H2O(g) ΔH°rxn = +44.01 kJ. Estimate the mass of water that must evaporate from the skin to cool the body by 0.50°C. Assume a body mass of 95 kg and assume that the specific heat capacity of the body is 4.0 J/g°C.

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134112831

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Calculate the total heat energy (q) required to cool the body by 0.50°C using the formula q = m * c * ΔT, where m is the mass of the body in grams, c is the specific heat capacity, and ΔT is the change in temperature.

Convert the body mass from kilograms to grams by multiplying by 1000, since 1 kg = 1000 g.

Substitute the values into the formula: m = 95000 g (since 95 kg = 95000 g), c = 4.0 J/g°C, and ΔT = 0.50°C.

Calculate the heat energy (q) in joules required to cool the body by 0.50°C.

Use the enthalpy change of the evaporation process (ΔH°rxn = +44.01 kJ/mol) to find the mass of water that must evaporate. Convert q from joules to kilojoules, then divide by ΔH°rxn to find the number of moles of water that need to evaporate. Finally, multiply by the molar mass of water (18.02 g/mol) to find the mass of water in grams.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Endothermic Process

An endothermic process is a chemical reaction or physical change that absorbs heat from its surroundings. In the context of evaporation, when water transitions from liquid to gas, it requires energy, which is taken from the skin, resulting in a cooling effect. This is quantified by the enthalpy change (ΔH°rxn), which indicates the amount of heat absorbed during the phase change.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For the human body, a specific heat capacity of 4.0 J/g°C means that it takes 4.0 joules of energy to increase the temperature of 1 gram of body mass by 1°C. This concept is crucial for calculating how much energy is needed to cool the body by a specific temperature change.

Mass-Energy Relationship

The mass-energy relationship in thermodynamics relates the mass of a substance to the energy required for a phase change or temperature change. In this scenario, the mass of water that must evaporate can be calculated using the formula Q = mcΔT, where Q is the heat absorbed, m is the mass of water, c is the specific heat capacity, and ΔT is the change in temperature. This relationship allows us to estimate the amount of water needed to achieve the desired cooling effect.

Related Practice

Textbook Question

Use standard enthalpies of formation to calculate the standard change in enthalpy for the melting of ice. (The ΔH°_f for H₂O(s) is –291.8 kJ/mol.) Use this value to calculate the mass of ice required to cool 355 mL of a beverage from room temperature (25.0 °C) to 0.0 °C. Assume that the specific heat capacity and density of the beverage are the same as those of water.

Dry ice is solid carbon dioxide. Instead of melting, solid carbon dioxide sublimes according to the equation: CO₂(s) → CO₂(g) ◀ When carbon dioxide sublimes, the gaseous CO₂ is cold enough to cause water vapor in the air to condense, forming fog. When dry ice is added to warm water, heat from the water causes the dry ice to sublime more quickly. The evaporating carbon dioxide produces a dense fog often used to create special effects. In a simple dry ice fog machine, dry ice is added to warm water in a Styrofoam cooler. The dry ice produces fog until it evaporates away, or until the water gets too cold to sublime the dry ice quickly enough. Suppose that a small Styrofoam cooler holds 15.0 L of water heated to 85 °C. Use standard enthalpies of formation to calculate the change in enthalpy for dry ice sublimation, and calculate the mass of dry ice that should be added to the water so that the dry ice completely sublimes away when the water reaches 25 °C. Assume no heat loss to the surroundings. (The ΔH°_f for CO₂(s) is –427.4 kJ/mol.)

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Textbook Question

Use standard enthalpies of formation to calculate the standard change in enthalpy for the melting of ice. (The ΔH°_f for H₂O(s) is –291.8 kJ/mol.)

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Textbook Question

In a sunny location, sunlight has a power density of about 1 kW/m². Photovoltaic solar cells can convert this power into electricity with 15% efficiency. If a typical home uses 385 kWh of electricity per month, how many square meters of solar cells are required to meet its energy requirements? Assume that electricity can be generated from the sunlight for 8 hours per day.

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Textbook Question

The kinetic energy of a rolling billiard ball is given by KE = 1/2 mv². Suppose a 0.17-kg billiard ball is rolling down a pool table with an initial speed of 4.5 m/s. As it travels, it loses some of its energy as heat. The ball slows down to 3.8 m/s and then collides head-on with a second billiard ball of equal mass. The first billiard ball completely stops and the second one rolls away with a velocity of 3.8 m/s. Assume the first billiard ball is the system. Calculate q.

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