Question

It has been determined that the body can generate 5500 \(\mathrm{kJ}\) of energy during one hour of strenuous exercise. Perspiration is the body's mechanism for eliminating this heat. What mass of water would have to be evaporated through perspiration to rid the body of the heat generated during 2 hours of exercise? (The heat of vaporization of water is 40.6 \(\mathrm{kJ} / \mathrm{mol.} )\)

Short answer

In the given problem, we have: Energy generated per hour = 5500 kJ/hour Duration of exercise = 2 hours Heat of vaporization of water = 40.6 kJ/mol Molar mass of water = 18.015 g/mol The total heat generated during the exercise is: Total heat generated = (5500 kJ/hour) × 2 hours = 11000 kJ Now, we can calculate the moles of water that must evaporate to remove the heat generated: moles of water = Total heat generated / Heat of vaporization of water moles of water = (11000 kJ) / (40.6 kJ/mol) ≈ 270.94 mol Finally, we can determine the mass of water corresponding to these moles: mass of water = moles of water × molar mass of water mass of water = (270.94 mol) × (18.015 g/mol) ≈ 4878.7 g Therefore, approximately 4878.7 g of water must be evaporated through perspiration to rid the body of the heat generated during 2 hours of exercise.

Step-by-step solution

Calculate the total heat generated in kJ during the exercise

The total heat generated can be found by multiplying the energy generated per hour by the number of hours the exercise lasts. In this case, 5500 kJ is generated each hour, and the exercise lasts for 2 hours. So the total heat generated is: Total heat generated = (5500 kJ/hour) × 2 hours

Determine the number of moles of water that must evaporate to remove the heat generated

To find the number of moles of water that must evaporate to eliminate the heat generated, use the formula: moles of water = Total heat generated / Heat of vaporization of water We already know the total heat generated and the heat of vaporization of water (40.6 kJ/mol), so we can plug those values into the formula to find the moles of water: moles of water = (Total heat generated) / (40.6 kJ/mol)

Calculate the mass of water that corresponds to the moles of water

Now that we have the number of moles of water, we can find the mass of water that corresponds to those moles. The molar mass of water is 18.015 g/mol. To find the mass of water, multiply the number of moles of water by the molar mass of water: mass of water = moles of water × molar mass of water Plug in the values for moles of water and the molar mass of water to find the mass of water: mass of water = (moles of water) × (18.015 g/mol) Now, you can calculate the mass of water that has to evaporate to eliminate the heat generated by the body during 2 hours of exercise.

Key concepts

Heat of vaporization

Understanding the concept of heat of vaporization is crucial when considering how the body cools itself during physical activities. The heat of vaporization refers to the amount of energy required to convert a specific amount of liquid into vapor without changing its temperature.

In the context of perspiration, when the body heats up during exercise, it needs to dispose of excess heat to maintain optimal internal temperatures. The body does this through sweating.

• As sweat evaporates from the skin, it absorbs a significant amount of heat.
• This cooling effect is due to the heat of vaporization of water.
• For water, this heat of vaporization is typically 40.6 kJ/mol.

For this process to be efficient, the body uses the energy transfer associated with water transitioning from a liquid to a vapor state. When you perspire, this allows the body to lose excess heat through the high energy required for this phase change.

Energy balance

Energy balance reflects the equilibrium the body must maintain between the energy it produces and the energy it dissipates. During exercise, the body generates a significant amount of heat energy, which must be adequately removed to prevent overheating. The concept of energy balance helps us understand how much energy needs to be dissipated to maintain stable internal temperatures.

In scenarios like this textbook exercise:

• The body generates 5500 kJ of energy per hour through intense physical activity.
• Over two hours, this amounts to a total energetic output of 11000 kJ.
• This energy must be balanced by the same amount being lost, in this case, through heat dissipation via perspiration.

The body achieves this balance by using perspiration as a cooling mechanism, leveraging the energy-intensive process of evaporating sweat to offset heat produced. This ensures that all the energy created is methodically released as perspiration evaporates.

Molar mass calculation

Molar mass, a fundamental chemical property, assists in converting moles of a substance to its mass in grams. This calculation is essential when determining how much of a substance, like water in sweat, is involved in reacting to or undergoing phase changes.

Here's how it applies to the exercise scenario:

• First, we find how many moles of water must evaporate using its heat of vaporization.
• Once the number of moles is determined, we calculate the actual mass of water needed for evaporation.
• The molar mass of water is 18.015 g/mol.
• Therefore, by multiplying the moles of water by this molar mass, you achieve the mass of water which must evaporate to eliminate 11000 kJ of heat.

This calculation reveals the quantity of sweat required, demonstrating the role molar mass plays in translating between chemical moles and physical mass, integral to energy balance in biological thermodynamics.