Problem 40 - College Physics

Question

\(\bullet\) Bicycling on a warm day. If the air temperature is the same as the temperature of your skin (about \(30^{\circ} \mathrm{C}\) ), your body cannot get rid of heat by transferring it to the air. In that case, it gets rid of the heat by evaporating water (sweat). During bicycling, a typical 70 kg person's body produces energy at a rate of about 500 \(\mathrm{W}\) due to metabolism, 80\(\%\) of which is converted to heat. (a) How many kilograms of water must the person's body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg}\) (b) The evaporated water must, of course, be replenished, or the person will dehydrate. How many 750 \(\mathrm{mL}\) bottles of water must the bicyclist drink per hour to replenish the lost water? (Recall that the mass of a liter of water is 1.0 kg.)

Step-by-Step Solution

Answer

The bicyclist needs to evaporate 0.595 kg of water, requiring 1 bottle (750 mL) to replenish it.

Step 1: Calculate the heat converted from metabolism

The total energy produced by a person's body during bicycling is 500 W. However, only 80% of this energy is converted into heat. To find the actual heat production, calculate:\[ Q_{heat} = 0.80 \times 500 \, \text{W} = 400 \, \text{W}. \]Since 1 W = 1 J/s, in one hour (3600 seconds), the heat energy generated is:\[ Q_{total} = 400 \, \text{W} \times 3600 \, \text{s} = 1,440,000 \, \text{J}. \]

Step 2: Determine the mass of water evaporated

Use the formula for heat transfer by vaporizing water: \[ \Delta Q = m \cdot L_v, \]where \( m \) is the mass of water in kg, \( \Delta Q \) is the heat to be eliminated (1,440,000 J), and \( L_v \) is the heat of vaporization (\(2.42 \times 10^6 \, \text{J/kg}\)). Solve for \( m \):\[ m = \frac{\Delta Q}{L_v} = \frac{1,440,000 \, \text{J}}{2.42 \times 10^6 \, \text{J/kg}} \approx 0.595 \, \text{kg}. \]

Step 3: Calculate the number of water bottles required

To replenish the evaporated water, convert the mass of water needed to volume. Knowing the density of water is 1 kg/L, 0.595 kg of water is 0.595 L. Each bottle is 0.75 L. Calculate the number of bottles required:\[ \text{Number of bottles} = \frac{0.595 \, \text{L}}{0.75 \, \text{L/bottle}} \approx 0.793. \]Since a person cannot drink a fraction of a bottle, round up to the nearest whole number: 1 bottle.

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy from a hotter object to a cooler one. In thermodynamics, it plays a crucial role in maintaining temperature balance in various systems, including the human body.
During physical activities like bicycling, the body's temperature rises, necessitating heat dissipation to prevent overheating. Ideally, our body can transfer heat through conduction to colder surrounding air. However, on warm days when the air temperature matches skin temperature, conduction becomes ineffective.
Consequently, the body resorts to other mechanisms such as evaporation to release excess heat. This highlights the importance of heat transfer in maintaining homeostasis, particularly in warm environments.

Metabolism

Metabolism refers to the biochemical processes that occur within a living organism to sustain life. It involves the conversion of food and drink into energy. While riding a bicycle, metabolism increases to meet the energy demands.
A significant portion of this energy converts into heat. As per the given example, a 70 kg person's body produces about 500 watts of energy through metabolism during cycling, with 80% transforming into heat. This results in 400 watts of heat generation, equivalent to 1,440,000 joules over one hour.
Understanding metabolism helps one grasp how physical activities impact energy production and heat generation in the human body.

Evaporation

Evaporation is the transition of a substance from a liquid to a gas. It is a critical process for heat dissipation, especially when conventional heat transfer modes fail.
The human body utilizes evaporation to regulate temperature, primarily through sweating. As sweat evaporates from the skin's surface, it draws heat away, cooling the body. During strenuous activities, the rate of evaporation must increase to combat the generated heat.
For instance, in our example, the body must efficiently evaporate water to expel 1,440,000 joules of heat. Understanding evaporation's role in thermoregulation emphasizes the need for adequate hydration to prevent dehydration.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a liquid into vapor without changing its temperature. This concept is vital in understanding how evaporation aids in heat dissipation.
In the cycling context, water's heat of vaporization at body temperature is approximately 2.42 million joules per kilogram. To eliminate 1,440,000 joules of heat, around 0.595 kg of water needs to be evaporated.
This reinforces the crucial role the heat of vaporization plays in energy balance and temperature regulation during physical activities. For continuous activity, replenishing the evaporated water becomes essential to maintain hydration levels.