91Ó°ÊÓ

While running, a \(70-\mathrm{kg}\) student generates thermal energy at a rate of 1200 \(\mathrm{W}\) . To maintain a constant body temperature of \(37^{\circ} \mathrm{C},\) this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the heat could not flow out of the student's body, for what amount of time could a student run before irreversible body damage occurred? (Note: Protein structures in the body are irreversibly damaged if body temperature rises to \(44^{\circ} \mathrm{C}\) or higher. The specific heat of a typical human body is 3480 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) , shightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)

Step by step solution

01

Determine the Temperature Increase

Compute the increase in temperature that leads to irreversible body damage. The initial safe temperature is \(37^{\circ} \mathrm{C}\) and damage occurs at \(44^{\circ} \mathrm{C}\). The temperature increase is \( \Delta T = 44^{\circ} \mathrm{C} - 37^{\circ} \mathrm{C} = 7^{\circ} \mathrm{C} = 7 \ \mathrm{K}\).

02

Calculate the Heat Needed for Temperature Increase

Use the formula for heat transfer: \( Q = mc\Delta T \), where \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.\( Q = 70 \ \mathrm{kg} \times 3480 \ \mathrm{J/kg \cdot K} \times 7 \ \mathrm{K} = 1,704,600 \ \mathrm{J} \).

03

Determine Time from Power and Heat

The student generates thermal energy at a power rate of \( 1200 \ \mathrm{W} \), which is equivalent to \( 1200 \ \mathrm{J/s} \). We need to calculate how long it takes to reach the total heat \( Q \) calculated previously.\( t = \frac{Q}{P} = \frac{1,704,600 \ \mathrm{J}}{1200 \ \mathrm{J/s}} = 1420.5 \ \mathrm{s} \).

04

Convert Time to Minutes

Convert the time from seconds to minutes for better intuition.\( t = \frac{1420.5}{60} \approx 23.68 \ \text{minutes}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Specific Heat Capacity

Specific heat capacity is a measure of how much thermal energy is needed to change the temperature of a specific mass by one degree Celsius or Kelvin. This concept is crucial when understanding how much heat a human body can absorb before its temperature changes. In the problem, the specific heat capacity of the human body is given as 3480 J/kgâ‹…K, which is slightly less than that of water due to the presence of proteins, fats, and minerals.
This means each kilogram of a human's body mass requires 3480 Joules of heat to increase in temperature by one Kelvin. Knowing this allows us to calculate how much heat is needed to raise our body's temperature by a certain amount, which can help keep us safe by predicting the limits before damage occurs.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hot object to a cooler one. In the human body, heat needs to be consistently eliminated to maintain a stable internal temperature, especially while engaging in activities like running. The body has mechanisms such as perspiration, breathing, and circulation that help to transfer excessive heat to the environment.
When these mechanisms fail or are overwhelmed, the body temperature can rise rapidly. Understanding how heat transfers in and out of the body helps us manage and mitigate potential overheating during physical exertion. It is important because excessive heat retention can potentially lead to damage.

Irreversible Body Damage

Irreversible body damage occurs when the body's temperature increases to a point where proteins and other cellular structures begin to break down. For humans, this temperature is approximately 44°C. At this stage, the body's proteins start to denature, meaning they lose their shape and functionality.
This damage disrupts normal physiological processes and can lead to severe outcomes, including heat stroke or damage to organs. Understanding the threshold for irreversible damage is crucial for managing risk during intense activities to avoid pushing the body beyond its safe operational limits.

Temperature Increase

Temperature increase in a system refers to the rise in thermal energy, leading to higher kinetic energy of molecules and consequently a higher temperature. For a human exercising, this means generating heat faster than it can be dissipated.
In the problem scenario, the temperature increase is calculated as a difference between the safe temperature (37°C) and the dangerous threshold (44°C). By understanding this increase, we can determine the amount of heat needed to reach this point and manage how quickly it could happen if heat is not being transferred out efficiently. Recognizing how temperature increases affect human physiology aids in safely planning and managing health under stress.