Problem 42 - University Physics with Modern Physics

Question

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 -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 \(\mathrm{kg.}\)

Step-by-Step Solution

Answer

(a) 0.595 kg of water, (b) 1 bottle of 750 mL water per hour.

Step 1: Calculate Heat Produced by the Body

First, determine the total energy converted to heat. If 80% of the 500 W produced is converted to heat, calculate the amount of heat produced: \[ 0.80 \times 500 \, \text{W} = 400 \, \text{W}. \]Then, convert this to total energy in one hour:\[ 400 \, \text{W} \times 3600 \, \text{s/hr} = 1,440,000 \, \text{J}. \]

Step 2: Calculate Mass of Water to Evaporate

Use the heat of vaporization to determine how much water needs to be evaporated to dissipate 1,440,000 J of heat. The heat of vaporization is given as \(2.42 \times 10^6 \, \text{J/kg}\). Set up the equation:\[ \text{Mass of water} = \frac{1,440,000 \, \text{J}}{2.42 \times 10^6 \, \text{J/kg}}. \]Solve for the mass of water:\[ \text{Mass of water} = 0.595 \text{ kg}. \]

Step 3: Convert Mass of Water to Number of Bottles

Determine how many 750 mL water bottles are needed. First, recall that 1 liter of water is 1 kg, so 750 mL of water is 0.75 kg. Divide the total mass of required water by the mass of one bottle:\[ \text{Number of bottles} = \frac{0.595 \, \text{kg}}{0.75 \, \text{kg/bottle}}. \]Calculate to find out how many bottles are necessary:\[ \text{Number of bottles} = 0.7933. \]

Step 4: Round Number of Bottles

Since you cannot consume a fraction of a bottle, round up to the nearest whole number. Thus, the bicyclist needs 1 bottle per hour to replenish the lost water.

Key Concepts

Evaporation

Evaporation is a crucial process in thermoregulation, especially on warm days. When your body cannot release heat into the air because it matches your skin’s temperature, it relies on evaporating sweat to cool down. As sweat evaporates, it absorbs heat from your skin, carrying it away in the process. This helps lower your body temperature.

However, evaporation can only occur if the surrounding air allows it, meaning it isn't already saturated with moisture. This is why on humid days, people often feel hotter because sweat doesn’t evaporate as efficiently, leading to less heat loss. In terms of thermoregulation, this process is vital as it helps maintain a stable body temperature in varying external conditions.

Metabolism

Metabolism is the set of life-sustaining chemical reactions in organisms. For humans, a significant part of our metabolic processes involves converting food into energy.

During activities like bicycling, metabolism ramps up to meet the energy demands of muscles. In the given exercise context, a 70 kg person produces energy at a rate of about 500 watts while cycling.

Notably, about 80% of this energy turns into heat rather than mechanical energy for movement.

• This heat needs to be dissipated to prevent overheating, which is where evaporation of sweat plays a key role.
• This is a natural response and an example of how metabolism interacts with thermoregulation to maintain body harmony.

Thus, understanding metabolism highlights its central role in body temperature regulation during physical exertion.

Heat of Vaporization

The heat of vaporization is the energy required to turn a liquid into a gas without changing its temperature. For water, this value stands at approximately 2.42 million joules per kilogram at body temperature.

This significant amount of energy is utilized during sweating. As water in sweat evaporates, it absorbs this heat from the body. This energy absorption is critical to cooling the body effectively.

• In the context of our exercise, the cyclist needs to evaporate a specified mass of water to dissipate heat produced during cycling.
• By calculating the exact amount of heat and relating it to the heat of vaporization value, one can determine how much water needs to evaporate.

This principle illustrates how the high heat of vaporization for water makes sweating an efficient thermoregulatory response.

Dehydration

Dehydration occurs when the body loses more water than it takes in, often through evaporation when sweating. This can be especially problematic during prolonged physical activity, leading to impaired performance and even health risks.

In the cycling scenario, the body's reliance on sweat for cooling means significant water loss that must be replenished.

• The exercise calculation shows that a cyclist would need to evaporate about 0.595 kg of water per hour.
• Thus, it becomes crucial to drink sufficient water, approximately equivalent to one 750 mL bottle per hour, to counteract this loss.

Ensuring adequate hydration is vital because even mild dehydration can reduce performance, increase heart rate, and strain the body's cooling mechanisms.

In essence, understanding dehydration's role in thermoregulation emphasizes the importance of maintaining water intake during and after exercise.