Problem 52 - University Physics with Modern Physics

Question

A person with skin of surface area 1.85 m\(^2\) and temperature 30.0\(^\circ\)C is resting in an insulated room where the ambient air temperature is 20.0\(^\circ\)C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second? (Recall that the room radiates back into the person and that the emissivity of the skin is 1.00.)

Step-by-Step Solution

Answer

The entropy of the air increases by the rate determined by \( \Delta S = \frac{P}{T_2} \) using the radiated power.

Step 1: Understand the Concept of Radiation

When a person is resting in a room with an air temperature lower than their skin temperature, they lose heat through radiation. According to the Stefan-Boltzmann law, the power radiated by a blackbody is proportional to the fourth power of its absolute temperature.

Step 2: Use the Stefan-Boltzmann Law

The Stefan-Boltzmann law is given by the equation \( P = \sigma \epsilon A (T_1^4 - T_2^4) \), where \( P \) is the power radiated, \( \sigma = 5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4 \) is the Stefan-Boltzmann constant, \( \epsilon = 1.00 \) is the emissivity, \( A = 1.85 \text{ m}^2 \) is the surface area, \( T_1 = 303 \text{ K} \) (person's skin temperature), and \( T_2 = 293 \text{ K} \) (room temperature).

Step 3: Calculate the Radiated Power

Substitute the values into the Stefan-Boltzmann equation: \( P = 5.67 \times 10^{-8} \times 1 \times 1.85 (303^4 - 293^4) \). Compute the result to get the power in watts.

Step 4: Calculate the Change in Entropy of the Air

The change in entropy \( \Delta S \) of the air can be calculated using the formula \( \Delta S = \frac{P}{T_2} \) where \( P \) is the power radiated and \( T_2 = 293 \text{ K} \). Solve it using the result from Step 3.

Key Concepts

Radiation

Radiation is a fundamental form of heat transfer that occurs through electromagnetic waves. This process does not require a medium, meaning it can even happen in a vacuum. When considering humans, we radiate heat energy into our surroundings naturally.

• Blackbody Radiation: A perfect emitter, known as a blackbody, emits the maximum possible radiation at a given temperature. Human skin is not a perfect blackbody, but for ease of calculations, we often use it as such in physics exercises. In these cases, skin emissivity (\(\epsilon\)) is considered to be 1.00.
• Stefan-Boltzmann Law: This law helps compute the power radiated by a body, given by the equation \(P = \sigma \epsilon A (T_1^4 - T_2^4)\). Here,
  • \(\sigma = 5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4\)
  • \(A\) is the surface area of the body
  • \(T_1\) and \(T_2\) are the absolute temperatures of the body and surroundings respectively.

The difference \(T_1^4 - T_2^4\) accounts for the net heat loss due to the body radiating more than it receives from its cooler surroundings.

Entropy Change

Entropy is a measure of disorder or randomness in a system and is a central concept in thermodynamics.

When heat flows from a hot body to a cooler one, the total entropy of the system increases, as heat energy disperses.

• Calculating Entropy Change: The change in entropy (\(\Delta S\)) is determined using the power (\(P\)) and the temperature of the surroundings (\(T_2\)). The formula is \(\Delta S = \frac{P}{T_2}\). Here,
  • \(P\) is the power calculated using the Stefan-Boltzmann Law.
  • \(T_2\) is the temperature of the surroundings in Kelvin, ensuring the units are consistent with the power value.

Entropy change is key to understanding how systems reach equilibrium, enhancing our grasp of energy efficiency and natural processes.

Heat Transfer

Heat transfer is a process where energy in the form of heat moves from a warmer object to a cooler one.

There are three main mechanisms for heat transfer: conduction, convection, and radiation. However, when it comes to radiation, this transfer happens through electromagnetic waves.

• Methods of Heat Transfer: While conduction requires direct contact between materials, convection involves the movement of fluid or gas. Radiation, on the other hand, can happen without any medium. It’s why we feel sun's heat even across space.
• Real-Life Application: Humans constantly exchange heat via radiation with their surroundings. It explains how we can feel our body warmth even in a room that is cooler than our body temperature.

In the exercise context, understanding these three modes of heat transfer is essential to appreciate how we naturally cool down in an environment where the air temperature is lower than our body temperature.