Problem 77 - College Physics

Question

". "The Ship of the Desert." Camels require very little water because they are able to tolerate relatively large changes in their body temperature. While humans keep their body temperatures constant to within one or two Celsius degrees, a dehydrated camel permits its body temperature to drop to \(34.0^{\circ} \mathrm{C}\) overnight and rise to \(40.0^{\circ} \mathrm{C}\) during the day. To see how effective this mechanism is for saving water, calculate how many liters of water a 400 -kg camel would have to drink if it attempted to keep its body temperature at a constant \(34.0^{\circ} \mathrm{C}\) by evaporation of sweat during the day \((12\) hours) instead of letting it rise to \(40.0^{\circ} \mathrm{C}\) . (Note: The specific heat of a camel or other mammal is about the same as that of a typical human, 3480 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) .\) The heat of vaporization of water at \(34^{\circ} \mathrm{C}\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} . )\)

Step-by-Step Solution

Answer

The camel needs approximately 40 liters of water.

Step 1: Understand the problem

We need to find how much water a camel would have to evaporate to maintain a constant body temperature of \(34.0^{\circ}C\) instead of letting it rise to \(40.0^{\circ}C\), using the known values of specific heat and heat of vaporization of water.

Step 2: Calculate heat gain

Determine the amount of heat the camel gains when its temperature rises from \(34.0^{\circ}C\) to \(40.0^{\circ}C\). Use the formula for heat gained, \(Q = mc\Delta T\), where \(m = 400 \text{ kg}\), \(c = 3480 \text{ J/(kg} \cdot \text{K)}\), and \(\Delta T = 40.0^{\circ}C - 34.0^{\circ}C = 6.0^{\circ}C\).

Step 3: Apply the formula for heat gained

Substitute the values into the formula: \[ Q = 400 \times 3480 \times 6.0 \]Calculate \(Q\).

Step 4: Calculate water needed for evaporation

Use the heat of vaporization of water at \(34.0^{\circ}C\), which is \(2.42 \times 10^{6} \text{ J/kg}\). Determine the mass \(m_{water}\) of the water that needs to evaporate:\[ m_{water} = \frac{Q}{L_v} \]where \(L_v = 2.42 \times 10^{6} \text{ J/kg}\) is the heat of vaporization.

Step 5: Compute the mass of water

Substitute the calculated heat gain \(Q\) and \(L_v\) into the formula to find \(m_{water}\).

Step 6: Convert mass of water to volume

Since the density of water is approximately \(1 \text{ kg/L}\), the mass in kilograms is numerically equal to the volume in liters. Therefore, the volume of water is the same number as \(m_{water}\).

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics and plays a critical role in many biological processes. It describes the movement of thermal energy from one object or substance to another, occurring in three primary ways: conduction, convection, and radiation. In biology, especially for endothermic organisms like camels, heat transfer can be essential to maintaining homeostasis.

• Conduction: Direct transfer of heat between molecules in contact. In living organisms, this occurs across tissues and through fluid flows.
• Convection: Transfer of heat by the physical movement of fluid, like blood, which helps distribute heat throughout the body.
• Radiation: Transfer of heat through electromagnetic waves. It is key in losing excess body heat to the environment.

In the camel's case, letting its body temperature rise and fall with ambient temperatures reduces the need for heat expulsion during the hottest parts of the day, conserving water normally lost through sweating.

Specific Heat Capacity

Specific heat capacity is a property that defines how much heat energy is needed to change the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). This concept is especially relevant in biology and helps explain why camels can endure drastic temperature changes.
For a camel, the specific heat capacity is similar to that of humans, about 3480 J/(kg·K). This tells us that it requires 3480 joules of energy to raise the temperature of 1 kilogram of camel tissue by 1 Kelvin. This high specific heat capacity allows camels to absorb more heat during the day without drastically raising their medium's temperature.

• Why it matters: A high specific heat capacity can act as a buffer against rapid temperature changes, aiding in thermal regulation.
• Biological Advantage: With more body mass comes a slower change in temperature, offering camels a natural thermal stability.

Such adaptations are crucial for survival in harsh, arid climates, where the temperature can fluctuate significantly between day and night.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a substance from a liquid to a vapor without a temperature change. For camels, understanding this concept helps explain how evaporative cooling is a vital adaptation for surviving in hot climates, limiting water loss through sweating.
At 34°C, the heat of vaporization for water is approximately 2.42 x 10^6 J/kg. This means to vaporize 1 kilogram of water at this temperature, 2.42 million joules of energy are necessary. It's the reason why sweating, a common cooling mechanism, is energy-intensive but effective.

• Biological Implication: Evaporating water through sweat removes heat from the body, substantially cooling the organism.
• Efficient Cooling: A small amount of sweat evaporating can dissipate a large amount of heat, critical for desert animals like camels.

In the exercise, calculating the heat of vaporization helps determine how much water a camel would need to maintain a constant temperature of 34°C instead of allowing a natural rise to 40°C.

Evaporation Cooling

Evaporation cooling is a process where liquid water is converted into vapor, absorbing significant heat in the process and thereby cooling the surface from which it evaporates. This principle is fundamental for thermoregulation in many species, including camels.
As camels sweat, water on their skin evaporates, taking with it a substantial amount of thermal energy. This process effectively cools the camel, which is particularly vital in environments where temperatures soar during the daytime.

• Efficiency: Evaporation uses less water compared to maintaining constant body temperatures through other heat dissipation methods like panting.
• Adaptation: By allowing body temperature to fluctuate, camels reduce the need for continuous sweating, conserving water.

Understanding evaporation cooling not only helps to explain the physiological adaptations of camels but also highlights the critical balance organisms must maintain between hydration and temperature regulation in extreme environments.