Problem 34 - University Physics with Modern Physics

Question

An average sleeping person metabolizes at a rate of about 80 \(W\) by digesting food or burning fat. Typically, 20% of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is 37\(^\circ\)C, and the skin is typically 7 C\(^\circ\) cooler. By how much does the person's entropy change per second due to this heat transfer?

Step-by-Step Solution

Answer

The entropy change per second is approximately 0.211 J/K.

Step 1: Understanding the Problem

The problem requires us to calculate the change in entropy for a sleeping person who transfers heat to the skin at a temperature of 30°C (since the internal temperature is 37°C and the skin is typically 7°C cooler). The metabolism rate is given, and we know that 80% of it translates into heat transfer. We should express temperatures in Kelvin for entropy calculations.

Step 2: Identify Energy Transfer

The metabolism rate of a sleeping person is 80 W, of which 80% becomes excess heat: \(80 \times 0.8 = 64\) W. This indicates the rate at which heat is being transferred from the person's body.

Step 3: Convert Temperature to Kelvin

The skin temperature in Celsius is 30°C (37°C - 7°C). Convert this to Kelvin: \(T_{skin} = 30 + 273.15 = 303.15\) K.

Step 4: Calculate Heat Transfer in Terms of Entropy

The change in entropy \(\Delta S\) per second due to heat transfer can be calculated using: \[\Delta S = \frac{Q}{T} \times \Delta t\]where \(Q\) is the heat transfer rate (64 J/s), \(T\) is the temperature of the skin in Kelvin (303.15 K), and \(\Delta t\) is one second.

Step 5: Perform the Entropy Change Calculation

Substitute known values into the entropy change formula:\[\Delta S = \frac{64}{303.15} \times 1 = 0.211 \text{ J/K}\]This is the rate of entropy change per second for the heat transfer from the body to the skin.

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept that occurs when thermal energy is exchanged between physical systems. In the context of the provided exercise, it involves the movement of heat from a person's body to their skin. This process is essential for regulating body temperature and maintaining homeostasis.
In a sleeping person, 80% of the metabolism energy is converted into heat. This means that if the total metabolic rate is 80 watts, 64 watts of energy is transferred as heat to the skin. This heat is then dissipated into the surrounding environment, primarily by thermal radiation.
Understanding heat transfer in this way helps us appreciate how our bodies naturally manage to stay cool even when generating significant amounts of heat. This heat transfer is crucial to keeping our body at an optimal operating temperature.

Metabolism

Metabolism is the set of life-sustaining chemical reactions that occur within our bodies. It helps convert food into energy that powers bodily functions, from basic cellular processes to physical movement.
In the exercise example, even while sleeping, a person metabolizes at a rate of about 80 watts. This metabolic rate reflects the energy the body utilizes per second to keep vital functions going, such as cell repair and blood circulation. Not all energy from metabolism is used directly; a substantial portion is converted into heat.
This heat is what the body needs to manage beyond basic physiological tasks, and managing this heat through transfer to the skin is part of the body's natural cooling system. Also, these processes illustrate how metabolism not only provides energy but also influences thermal regulation.

Thermal Radiation

Thermal radiation is a mode of heat transfer that involves the emission of electromagnetic waves. It occurs when the body's surface emits infrared radiation, allowing heat to dissipate into the surrounding environment.
The exercise highlights this process as the primary way a person's excess metabolic heat is released. As heat moves to the skin, it radiates into the air, reducing the body's internal heat load.
Understanding thermal radiation is vital as it plays a key role in how living organisms regulate temperature. This natural mechanism does not require direct contact with other objects or conductors, making it an effective and efficient means of heat transfer.

Temperature Conversion

Temperature conversion is an important aspect when dealing with thermodynamic calculations, like the entropy change due to heat transfer. In the exercise, we need to convert temperature from Celsius to Kelvin to use in entropy calculations.
Kelvin is the SI unit for temperature used in scientific calculations, especially when dealing with formulas that involve thermodynamics. The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature.
In our example, the skin temperature at 30°C converts to 303.15 Kelvin. By using these converted temperatures, we can accurately apply thermodynamic equations to determine the entropy change due to heat transfer. This ensures standardized results across various scientific disciplines.