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Chapter 17: Problem 31

While running, a 70 -kg student generates thermal energy at a rate of 1200 \(\mathrm{W}\) . For the runner 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 \(3480 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K},\) 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 could run for approximately 23.66 minutes before risking body damage.

Step by step solution

01

Identify Key Information

We are given that the student has a mass of 70 kg, generates thermal energy at a rate of 1200 W, and the specific heat capacity of the human body is 3480 J/kg*K. The body temperature rises from 37掳C to 44掳C, giving a temperature change (螖T) of 7掳C or 7 K.
02

Calculate Total Heat

Use the formula for heat, which is \( Q = m \cdot c \cdot \Delta T \), where \( Q \) is the heat in Joules, \( m \) is the mass in kg, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change. Substituting the given values:\[Q = 70 \text{ kg} \times 3480 \text{ J/kg} \cdot \text{K} \times 7 \text{ K}\]\[Q = 1,703,600 \text{ J}\]
03

Calculate Time

The power generated is 1200 W, which means 1200 J of energy is produced every second. To find the time \( t \) it would take to reach the critical temperature if the heat isn't removed, use:\[t = \frac{Q}{P}\]where \( P \) is the power in watts.\[t = \frac{1,703,600 \text{ J}}{1200 \text{ W}}\]\[t = 1419.67 \text{ seconds}\]
04

Convert Time to Minutes

Since time is often more usefully expressed in minutes, divide by 60 to convert seconds to minutes:\[t = \frac{1419.67}{60} \approx 23.66 \text{ minutes}\]

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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 an essential concept in understanding how heat is transferred in the human body. It defines the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). For the human body, the specific heat capacity is about 3480 J/kg路K. This value is slightly less than water due to the presence of proteins, fats, and minerals, which have lower specific heat capacities.
Understanding this helps in calculating how much energy is stored in or released by the body when there is a change in temperature. For example, if a runner generates a lot of thermal energy, the specific heat capacity indicates how much this energy will affect the body鈥檚 temperature shift.
- **Relevance**: The higher the specific heat capacity, the more energy is required to change the temperature, and vice versa.- **Application**: It is used to calculate the total heat production in the body through the formula: \( Q = m \cdot c \cdot \Delta T \), where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the change in temperature.
Thermal Energy
Thermal energy in the human body refers to the total kinetic energy of molecules due to their random motions, often manifested as body heat. It is influenced by various factors such as metabolic rate, physical activity, and environmental conditions.
In our running example, the student produces thermal energy at a rate of 1200 W due to metabolic activity. If this energy isn鈥檛 expelled efficiently, it can lead to an unsafe rise in body temperature.
- **Where It Comes From**: Generated during muscle activity, such as running, and biochemical reactions like metabolism. - **Importance**: Excessive thermal energy without proper dissipation mechanisms like sweating can lead to overheating and potential damage. - **Calculation**: Calculations involving thermal energy use power (W), where power is the rate of energy transfer, integrating with time to determine the total heat produced over a session.
Body Temperature Regulation
Body temperature regulation is crucial for maintaining homeostasis. The human body uses various mechanisms to stabilize internal temperature, ensuring it stays within a safe range, typically around 37掳C. This regulation involves adjusting heat production and loss through mechanisms like sweating, blood flow adjustments, and metabolic control.
- **Homeostasis**: It is the body's tendency to maintain a steady internal state, balancing heat gain and loss. - **Mechanisms**: Sweating evaporates heat, blood vessels dilate to release heat, and metabolic rates adjust to generate or restrict more heat.
In the exercise example, if the body fails to lose excess heat (like when running generates 1200 W of thermal energy), the temperature can rise dangerously, reaching the point where proteins might denature (around 44掳C), causing irreversible damage.
Efficient body temperature regulation mechanisms are crucial in extreme scenarios. The concept emphasizes the body's need to maintain a delicate balance between energy generation and heat dissipation to avoid critical overheating.

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Most popular questions from this chapter

A wood ceiling with thermal resistance \(R_{1}\) is covered with a layer of insulation with thermal resistance \(R_{2} .\) Prove that the effective thermal resistance of the combination is \(R=R_{1}+R_{2}\) .

On-Demand Water Heaters. Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawbacks are that energy is wasted because the tank loses heat when it is not in use and that you can run out of hot water if you use too much. Some utility companies are encouraging the use of on-demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is 2.5 gal/min (9.46 L/min) with the tap water being heated from \(50^{\circ} \mathrm{F}\left(10^{\circ} \mathrm{C}\right)\) to \(120^{\circ} \mathrm{F}\left(49^{\circ} \mathrm{C}\right)\) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

The Sizes of Stars. 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 \(2.7 \times 10^{32} \mathrm{W}\) and has surface temperature \(11,000 \mathrm{K}\) ; (b) Procyon \(\mathrm{B}\) (visible only using a telescope), which radiates energy at a rate of \(2.1 \times 10^{23} \mathrm{W}\) and has surface temperature \(10,000 \mathrm{K}\) (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 \(\mathrm{B}\) is an example of a white dwarf star.)

(a) On January \(22,1943,\) the temperature in Spearfish, South Dakota, rose from \(-4.0^{\circ} \mathrm{F}\) to \(45.0^{\circ} \mathrm{F}\) in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was \(44.0^{\circ} \mathrm{F}\) on January \(23,1916 .\) The next day the temperature plummeted to \(-56^{\circ} \mathrm{F} .\) What was the temperature change in Celsius degrees?

Steam Burns Versus Water Burns. What is the amount of heat input to your skin when it receives the heat released (a) by 25.0 g of steam initially at \(100.0^{\circ} \mathrm{C},\) when it is cooled to skin temperature \(\left(34.0^{\circ} \mathrm{C}\right) ?\) (b) By 25.0 \(\mathrm{g}\) of water initially at \(100.0^{\circ} \mathrm{C},\) when it is cooled to \(34.0^{\circ} \mathrm{C} ?(\mathrm{c})\) What does this tell you about the relative severity of steam and hot water burns?
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