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BIO While running, a 70-kg student generates thermal energy at a rate of 1200 W. For the runner to maintain a constant body temperature of 37\(^\circ\)C, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the energy 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\(^\circ\)C or higher. The specific heat of a typical human body is 3480 J / kg \(\cdot\) K, slightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)

Short Answer

Expert verified
The student can run for about 23.6 minutes before body damage occurs.

Step by step solution

01

Identify the Known Values

First, list all the known values from the problem:- Mass \( m = 70 \text{ kg} \)- Power generated \( P = 1200 \text{ W} \)- Initial temperature \( T_i = 37\degree{C} \)- Final temperature \( T_f = 44\degree{C} \)- Specific heat \( c = 3480 \text{ J/kg} \cdot K \).
02

Determine Temperature Change

Calculate the change in temperature \( \Delta T \) using the temperature values:\[ \Delta T = T_f - T_i = 44\degree{C} - 37\degree{C} = 7\degree{C} \]
03

Calculate Energy Required for Temperature Change

Use the specific heat formula \( Q = mc\Delta T \) to calculate the energy required to raise the body's temperature:\[ Q = m \times c \times \Delta T = 70 \times 3480 \times 7 = 1,700,400 \text{ J} \]
04

Calculate Time Until Irreversible Damage

Determine the time \( t \) it takes for the energy to reach \( Q \) using power with \( Q = P \times t \):\[ t = \frac{Q}{P} = \frac{1,700,400}{1200} \approx 1417 \text{ seconds} \].Convert seconds to minutes for easier interpretation:\[ 1417 \text{ seconds} \div 60 \approx 23.6 \text{ minutes} \]
05

Conclusion

The student can run for approximately 23.6 minutes before the body temperature reaches the critical level that might cause irreversible damage.

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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 an essential concept in understanding how much heat energy it takes to change the temperature of a substance. Specifically, it refers to the amount of heat needed to raise the temperature of one kilogram of a material by one degree Celsius or Kelvin.

In this exercise, we observed that the specific heat capacity of the human body is 3480 J/kg·K. This means that to increase one kilogram of body mass by 1°C, 3480 Joules of energy are required. This value is close to but lower than the specific heat of water, due to the presence of proteins and fats which have additional thermal properties.

Understanding the specific heat capacity helps us determine how rapidly or slowly a body can heat up, which is crucial for athletes and people engaged in physical activities. The higher the specific heat, the more energy is needed to increase temperature, affecting how the human body absorbs and dissipates heat during various conditions.
Body Temperature Regulation
The human body has evolved with several mechanisms for maintaining its core temperature around 37°C, regardless of external conditions. This process, known as thermoregulation, involves balancing heat production with heat loss.

During physical activities such as running, the body generates more heat. To prevent overheating, it must eliminate this extra thermal energy through various methods like sweating or increased blood flow to the skin. These mechanisms help maintain the core body temperature within a safe range.

In the absence of these cooling methods, like in our exercise example, the body could potentially heat up to dangerous levels (44°C in this case). When this happens, proteins and enzymes may denature, leading to life-threatening conditions. Thus, the ability to regulate body temperature effectively is vital for health and performance.
Thermal Energy Transfer
Thermal energy transfer in biological systems occurs through various mechanisms to maintain a stable internal environment. Heat can move by conduction, convection, and radiation, all of which play roles in how the body cools down or warms up in different situations.

In the given scenario, thermal energy is generated at a rate of 1200 watts while running. If not efficiently dissipated, this energy could accumulate, raising body temperature towards the critical level.
The process of transferring heat away can include:
  • Perspiration: Sweat evaporates from the skin surface, a process that absorbs heat energy and cools the body.
  • Blood Flow: Increased circulation brings warmer blood to the cooler surface of the skin.
  • Breathing: Exhaling warmer air and inhaling cooler air aids in heat loss.
These processes rely on the effective exchange of thermal energy, ensuring the body does not overheat during intense activities.

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Most popular questions from this chapter

An aluminum tea kettle with mass 1.10 kg and containing 1.80 kg of water is placed on a stove. If no heat is lost to the surroundings, how much heat must be added to raise the temperature from 20.0\(^\circ\)C to 85.0\(^\circ\)C?

A 6.00-kg piece of solid copper metal at an initial temperature \(T\) is placed with 2.00 kg of ice that is initially at -20.0\(^\circ\)C. The ice is in an insulated container of negligible mass and no heat is exchanged with the surroundings. After thermal equilibrium is reached, there is 1.20 kg of ice and 0.80 kg of liquid water. What was the initial temperature of the piece of copper?

The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of \(2.7 \times 10{^3}{^2} W\) and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of \(2.1 \times 10{^2}{^3} W\) and has surface temperature 10,000 K. (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a supergiant star, and Procyon B is an example of a white dwarf star.)

BIO Conduction Through the Skin. The blood plays an important role in removing heat from the body by bringing this energy directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. Assume that the blood is brought to the bottom layer of skin at 37.0\(^\circ\)C and that the outer surface of the skin is at 30.0\(^\circ\)C. Skin varies in thickness from 0.50 mm to a few millimeters on the palms and soles, so assume an average thickness of 0.75 mm. A 165-lb, 6-ft-tall person has a surface area of about 2.0 m\(^2\) and loses heat at a net rate of 75 W while resting. On the basis of our assumptions, what is the thermal conductivity of this person’s skin?

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is 0.300 m and the length of the copper section is 0.800 m. Each segment has cross-sectional area 0.00500 m\(^2\). The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice–water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings. (a) What is the temperature of the point where the brass and copper segments are joined? (b) What mass of ice is melted in 5.00 min by the heat conducted by the composite rod?

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