Problem 53 - University Physics with Modern 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 \(400-\mathrm{kg}\) 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 would need to drink approximately 3.45 liters of water.

Step 1: Understand the Problem

The camel allows its body temperature to vary from \(34.0^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\). If we want to keep it constantly at \(34.0^{\circ} \mathrm{C}\), the camel would need to evaporate water (sweat) to dissipate the excess heat. The task is to calculate how much water would be required for this process.

Step 2: Calculate the Heat Required to Keep Constant Temperature

First, find the amount of thermal energy that the camel needs to dissipate if its temperature rises by \(6.0^{\circ} \mathrm{C}\) (from \(34.0^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\)). Use the formula: \[ Q = mc\Delta T \]where \( m = 400 \text{ kg} \) is the mass of the camel, \( c = 3480 \text{ J/kg} \cdot \text{K} \) is the specific heat capacity, and \( \Delta T = 6.0 \underline{\phantom{xxx}}^{\circ} \mathrm{C} \). Substitute these values:\[ Q = 400 \times 3480 \times 6 = 8,352,000 \text{ J} \]

Step 3: Calculate the Mass of Water Needed

Now, use the heat of vaporization for water to find the mass of water that needs to be evaporated to dissipate \(8,352,000 \text{ J}\) of heat. The formula is:\[ Q = m_v L_v \]where \( m_v \) is the mass of the evaporated water, and \( L_v = 2.42 \times 10^6 \text{ J/kg} \) is the heat of vaporization of water.Rearranging for \( m_v \):\[ m_v = \frac{Q}{L_v} = \frac{8,352,000}{2.42 \times 10^6} \approx 3.45 \text{ kg} \]

Step 4: Convert the Mass of Water to Liters

Since 1 liter of water has a mass of 1 kg, the camel would need approximately 3.45 kg of water, which is 3.45 liters.

Key Concepts

specific heat capacity

The specific heat capacity is an essential concept when understanding how substances respond to temperature changes. Specific heat capacity is the amount of heat energy required to raise the temperature of 1 kilogram of a substance by 1 degree Celsius (\(1 \text{ kg} \cdot \text{K}\)). It is a measure of a substance's thermal inertia or resistance to temperature change.

In the context of a camel, which has a similar specific heat capacity to humans at 3480 \(\text{J/km} \cdot \text{K}\), it measures how well camels can store and manage heat within their bodies.

• Higher specific heat capacity means the animal can absorb a lot of heat without a significant increase in temperature.
• This ability helps camels in hot environments to avoid overheating as their body can store more heat throughout the day without sweating.

This capability is vital in arid environments, as it greatly reduces water loss through sweat, which is crucial for survival in water-scarce regions.

Understanding specific heat capacity can offer insights into how animals regulate their internal temperatures, particularly in extreme environments, by efficiently managing heat energy without excessive water loss. This is critical in understanding thermoregulation strategies in different species.

evaporation and cooling

Evaporation is a key process in thermoregulation for many animals, including camels. When water changes from liquid to vapor, it takes in energy, which results in a cooling effect on the surface from which it evaporates.

This process is known as evaporative cooling and is essential in maintaining a stable internal temperature.

• For camels, if maintaining constant body temperature through sweating, the heat (\(Q\)) dissipated through evaporation equals the energy needed to increase the body temperature.
• The heat of vaporization of water at the camel's body temperature (\(34^{\circ} \mathrm{C}\)) is 2.42 × 10^6 \(\text{J/kg}\), representing the energy for each kilogram of water that evaporates to cool the body.

Through this method, camels could use significant amounts of water to shed excess heat if they did not allow their temperature to fluctuate.

The reliance on evaporative cooling demonstrates an elegant balance between temperature regulation and water conservation, a delicate trade-off managed by camels in desert climates.

camel physiology

Camel physiology is fascinating and well-adapted for survival in the harsh, arid desert environment. One remarkable feature is their ability to tolerate significant fluctuations in body temperature.

Camels can allow their body temperature to drop to \(34^{\circ} \mathrm{C}\) and rise to \(40^{\circ} \mathrm{C}\) in response to extremes in environmental temperature. This temperature range helps conserve water by reducing the need for evaporation through sweating.

• This physiological adaptation means camels don’t have to sweat excessively, conserving vital water resources.
• Camels also have other adaptations, such as thick fur insulating against heat and sandstorms while reducing water loss.

Their thick coats reflect sunlight and provide insulation, and their unique fat storage in humps also plays a role in water conservation, as metabolizing fat yields water.

By allowing body temperature to vary, camels maximize their scarce resources and adapt to environments where other animals might not survive. This efficient system goes beyond simple cooling; it is a holistic approach to thriving in one of the world’s most challenging climates.