Problem 89 - University Physics with Modern Physics

Question

If the air temperature is the same as the temperature of your skin (about 30\(^\circ\)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 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 \(2.42 \times 10{^6} J/kg\). (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 kg.)

Step-by-Step Solution

Answer

The cyclist needs to evaporate approximately 0.595 kg of water per hour and should drink at least one 750-mL bottle of water to replenish it.

Step 1: Calculate Total Heat Generated

The energy produced by the body is 500 W, but only 80% of this is converted to heat. To find the heat produced in one hour, first calculate the power that turns into heat: \( P_{heat} = 0.8 \times 500 \text{ W} = 400 \text{ W} \). Convert this into energy over one hour (3600 seconds): \( Q = P_{heat} \times 3600 \text{ s} = 400 \times 3600 = 1.44 \times 10^6 \text{ J} \).

Step 2: Calculate Mass of Water to Evaporate

To find how much water needs to evaporate to remove this heat, use the formula: \( Q = mL_v \), where \( L_v = 2.42 \times 10^6 \text{ J/kg} \) is the heat of vaporization. Solving for mass \( m \): \( m = \frac{Q}{L_v} = \frac{1.44 \times 10^6 \text{ J}}{2.42 \times 10^6 \text{ J/kg}} \approx 0.595 \text{ kg} \).

Step 3: Determine Number of Water Bottles Needed

Each 750-mL bottle is equivalent to 0.75 kg of water, since 1 L = 1 kg. To replace the 0.595 kg of water lost, calculate the number of bottles: \( \text{Number of bottles} = \frac{0.595}{0.75} \approx 0.793 \). Since only whole bottles can be consumed, the person must drink at least 1 full bottle to fully replenish the lost water.

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, dealing with the way heat moves from one object or substance to another. In our bodies, heat transfer typically occurs from the body to the surrounding environment, helping to maintain body temperature. However, when the air temperature matches our skin temperature, as in the exercise, the common method of cooling down by heat transfer to the air becomes ineffective. This situation can pose a challenge because our body generates heat continuously due to metabolic processes. Without an efficient way to transfer this excess heat to the surroundings, our body must rely on an alternative method to regulate temperature, leading us to the next concept: evaporation.

Evaporation

Evaporation is a key cooling mechanism for the human body, particularly when external heat transfer options are insufficient. It involves the transformation of liquid water, such as sweat, into vapor, a process that requires energy. This energy, known as the heat of vaporization, is absorbed from the body and surrounding tissues.

In the exercise, when bicycling, 80% of the energy from metabolism becomes heat that needs to be dissipated. Evaporation of sweat is a crucial way the body gets rid of this heat. The latent heat of vaporization for water is quite high, at approximately 2.42 million Joules per kilogram. This means evaporating even small amounts of water can carry away significant amounts of heat, which protects the body from overheating.

Metabolism

Metabolism refers to the chemical processes that occur within a living organism to sustain life. It includes the conversion of food into energy and is the source of ongoing heat production in the body. During physical activities, like cycling, the metabolic rate increases, leading to more energy production and consequently more heat.

In our example, a person cycling continuously generates power at a rate of 500 watts. Of this, 80%, or 400 watts, is converted into heat energy each hour, leading to the body having to find a way to remove this large amount of heat to maintain a stable internal environment.

Energy Conversion

Energy conversion in the context of the human body involves transforming stored chemical energy from food into other forms like kinetic and thermal energy. As the body engages in activities such as cycling, the muscle cells convert energy into motion and heat.

In the given problem, the body is using metabolic energy which is mostly converted into heat due to the effort of cycling. Since 400 watts of energy turns into heat, it poses a need for effective energy management via cooling strategies like evaporation. Here, energy conversion also highlights the importance of replenishing electrolytes and fluids lost through sweating, as maintaining hydration is vital for continued energy conversion and overall bodily function.