Question

Metabolic activity in the human body releases approximately \(1.0 \times 10^{4} \mathrm{~kJ}\) of heat per day. Assume that a \(55-\mathrm{kg}\) body has the same specific heat as water; how much would the body temperature rise if it were an isolated system? How much water must the body eliminate as perspiration to maintain the normal body temperature \(\left(98.6^{\circ} \mathrm{F}\right)\) ? Comment on your results. (The heat of vaporization of water is \(2.41 \mathrm{~kJ} / \mathrm{g}\).)

Short answer

Without heat loss, body temperature would rise to 177.3°F, requiring 4.149 kg of perspiration to maintain 98.6°F.

Step-by-step solution

Understanding the specific heat capacity

We begin by recognizing that the specific heat capacity of water is \[ c = 4.186 \text{ J/g}^\circ\mathrm{C}. \] Since 1 kg equals 1000 g, for a 55 kg body, we have \[ m = 55,000 \text{ g}. \]

Calculate the temperature change without heat loss

The change in body temperature, \( \Delta T \), can be calculated using \[ Q = mc\Delta T, \] where \( Q = 10,000 \text{ kJ} = 10,000,000 \text{ J} \) (since 1 kJ = 1000 J). Solving for \( \Delta T \), we get: \[ \Delta T = \frac{Q}{mc} = \frac{10,000,000}{55,000 \times 4.186} \approx 43.7^\circ\mathrm{C}. \]

Convert temperature rise to Fahrenheit

First, convert the initial temperature (98.6°F) to Celsius: \[ T_C = \frac{98.6 - 32}{1.8} \approx 37^\circ\mathrm{C}. \]Adding the temperature change \( \Delta T \), the new temperature in Celsius is: \[ T_{new,C} = 37 + 43.7 = 80.7^\circ\mathrm{C}. \]Convert back to Fahrenheit to find: \[ T_{new,F} = 80.7 \times 1.8 + 32 \approx 177.3^\circ\mathrm{F}. \]

Calculating water needed for perspiration

To find the amount of water needed to dissipate the heat through vaporization, use the formula \[ Q = m' \times L, \]where \( m' \) is the mass of perspired water and \( L = 2.41 \text{ kJ} / \text{g} = 2410 \text{ J/g} \). Given \( Q = 10,000,000 \text{ J} \), \[ m' = \frac{Q}{L} = \frac{10,000,000}{2,410} \approx 4,149 \text{ g}. \] This means approximately 4,149 grams or 4.149 kg of water needs to be perspired.

Conclusion: Interpreting the Results

Without evaporation to release the heat, body temperature would rise significantly, potentially to 177.3°F, which is unsustainable for human health. Therefore, the body must perspire approximately 4.149 kg of water to maintain a normal internal temperature of 98.6°F.

Key concepts

Specific Heat Capacity

Specific heat capacity is a crucial concept in understanding temperature changes in substances. It refers to the amount of heat required to change the temperature of a unit mass of a material by 1 degree Celsius. In our exercise, the specific heat capacity of the human body is assumed to be that of water, approximately 4.186 J/g°C.
This means it takes 4.186 joules to increase the temperature of 1 gram of water by 1°C. Understanding this helps us realize how stable our body temperature is despite daily activities. Our bodies have a large mass compared to daily temperature changes, which helps buffer against drastic temperature shifts due to metabolic heat production.

• The specific heat of water and the factors influencing a material's ability to absorb heat are related.
• Water's high specific heat capacity allows it to store large quantities of heat without experiencing significant temperature changes.

Heat of Vaporization

The heat of vaporization is another essential concept in thermal management of the human body. It refers to the amount of energy required to convert a unit mass of a substance from liquid to vapor without changing its temperature. In the context of the exercise, this applies to water being evaporated through perspiration to cool the body.
When water in sweat evaporates, it absorbs a large amount of heat from the body due to its high heat of vaporization, which is 2.41 kJ/g. This process effectively releases excess heat generated from metabolic activities, thus regulating body temperature.

• The high heat of vaporization of water ensures efficient cooling with minimal water loss.
• Evaporation involves breaking of intermolecular bonds, which requires substantial energy.
• Through this mechanism, a seemingly small mass of perspired water can have a significant cooling effect.

Temperature Conversion

Temperature conversion, especially between Fahrenheit and Celsius, is essential in many scientific contexts, including the study of metabolic heat regulation. The formula to convert Fahrenheit to Celsius is \( T_C = \frac{T_F - 32}{1.8} \), and to convert back from Celsius to Fahrenheit, use \( T_F = T_C \times 1.8 + 32 \).
In our exercise, converting from one temperature scale to another helped determine how much the body temperature rises in a scenario where heat is not lost. Understanding these conversions allows for full comprehension of temperature impacts in everyday and scientific settings.

• Understanding how to interchange between these scales is vital for interpreting results in thermodynamics.
• Mistakes in conversion could lead to incorrect results, impacting understanding and calculations significantly.
• The conversion formulas are simple yet fundamental for temperature-related studies.

Perspiration Cooling

Perspiration cooling is a natural method humans use to regulate body temperature. By evaporating perspiration, the body releases excess metabolic heat, maintaining appropriate internal conditions. This process highlights the importance of sweating, especially under high temperatures or physical exertion.
In the exercise, we calculated that approximately 4.149 kg of water needs to be evaporated to counterbalance the daily metabolic heat produced. This illustrates how efficient perspiration is as a cooling mechanism.

• Perspiration prevents overheating by utilizing the heat of vaporization effectively.
• Efficiency in cooling depends on factors like humidity and air movement, as these affect evaporation rates.
• Understanding perspiration cooling helps appreciate the body's natural ability to sustain temperature balance.