Question

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

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

Step-by-step solution

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 \( c = 3480 \text{ J/kg} \cdot K \).

Determine Temperature Change

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

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:\[ Q = m \times c \times \Delta T = 70 \times 3480 \times 7 = 1,700,400 \text{ J} \]

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:\[ 1417 \text{ seconds} \div 60 \approx 23.6 \text{ minutes} \]

Conclusion

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

Key concepts

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.