Problem 75 - College Physics

Question

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

A camel's body temperature changes from \(34^{\circ} C\) to \(40^{\circ} C\) during the day. We need to calculate how much water is required to maintain the camel's body temperature at \(34^{\circ} C\) instead.

Step 2: Calculate Heat Loss Needed to Keep Temperature Constant

We need to find out the amount of heat energy (in joules) that needs to be removed from the camel's body to prevent its temperature from increasing by \(6^{\circ} C\). Use the formula for heat change: \[ Q = mc\Delta T \] where \(m\) is the mass of the camel (\(400\, \text{kg}\)), \(c\) is the specific heat capacity (\(3480\, \text{J/(kg}\cdot\text{K)}\)), and \(\Delta T\) is the temperature change (\(6\, \text{K}\)).

Step 3: Solve for Total Heat Loss

Substitute the given values into the heat change formula to calculate \( Q \): \[ Q = 400 \, \text{kg} \times 3480 \, \text{J/(kg}\cdot\text{K)} \times 6 \, \text{K} \] Calculate \( Q \).

Step 4: Determine Water Needed for Evaporation

Using the calculated heat loss \(Q\), find out how much water needs to evaporate to dissipate this heat using the formula: \[ m_{\text{water}} = \frac{Q}{L_v} \] where \(L_v = 2.42 \times 10^6 \, \text{J/kg}\) is the latent heat of vaporization of water. Solve for \(m_{\text{water}}\).

Step 5: Convert Mass of Water to Volume

Convert the mass of water calculated into volume (liters). Use the density of water, which is approximately \(1 \, \text{kg/L}\). Thus, volume \(V\) of water is approximately equal to its mass \(m_{\text{water}}\).

Step 6: Final Calculation and Result

Using the previous calculations:\[ Q = 400 \, \text{kg} \times 3480 \, \text{J/(kg}\cdot\text{K)} \times 6 \, \text{K} = 8,352,000 \, \text{J} \]\[ m_{\text{water}} = \frac{8,352,000 \, \text{J}}{2.42 \times 10^{6} \, \text{J/kg}} \approx 3.45 \, \text{kg} \]Since water density is \(1 \, \text{kg/L}\), the camel would need approximately \(3.45 \, \text{L}\) of water to keep the temperature constant.

Key Concepts

Specific Heat Capacity

Specific heat capacity is an important property in thermoregulation, especially for living organisms. It refers to the amount of heat energy required to change the temperature of a unit mass of a substance by one degree Kelvin. For camels, as well as other mammals, the specific heat capacity is roughly similar to that of humans, which is around 3480 J/(kg·K).
This property tells us how well a camel can absorb heat. A higher specific heat capacity means the camel can take in more heat energy without a large increase in temperature. This is particularly useful for camels, enabling them to withstand harsh desert conditions during hot days. As camels have a relatively high specific heat capacity, they can tolerate temperature fluctuations without needing as much water to cool down compared to other animals.

Heat of Vaporization

Heat of vaporization is crucial for understanding how animals regulate their body temperature through processes like sweating. This property indicates the amount of heat energy needed to convert a unit mass of a liquid into vapor without a temperature change. For water, the latent heat of vaporization is approximately 2.42 x 10^6 J/kg.
In terms of camels, when their body temperature rises, evaporating sweat helps cool them down. This is because water absorbs a lot of heat when it turns into vapor, removing this energy from the camel's body. Calculations involving the heat of vaporization are used to determine the volume of water camels need to evaporate to cool themselves effectively, especially during extreme temperature increases.

Camel Physiology

Camels are fascinating creatures due to their unique physiological adaptations that allow them to survive in extreme climates with minimal water. One key trait is their ability to let their body temperature fluctuate significantly. While humans maintain a nearly constant body temperature, camels can let their core temperature drop by night and rise during the day.
This ability helps camels save water since they don't sweat to stay cool. Instead, they use other mechanisms like stored fat in their humps that can be metabolized for water and energy when food or water is scarce. Understanding camel physiology helps explain why they are such resilient animals capable of surviving harsh desert environments.

Temperature Regulation

Temperature regulation in animals is a vital process for maintaining a stable internal environment, allowing them to function effectively. For camels, this involves controlling their body temperature through methods other than continuous sweating or panting, which would lead to excessive water loss.
By letting their body temperature vary, camels avoid unnecessary water waste. When temperatures rise, they might use the heat during the day to store energy, which saves them from constant evaporation cooling. This regulation ensures they can save water for more critical situations, highlighting their evolution to thrive in arid conditions.

Water Conservation Strategies

Camels have a myriad of water conservation strategies that enable them to survive with minimal hydration in desert environments. First, as mentioned, they permit significant fluctuations in body temperature, which reduces the need to cool through sweating.
Furthermore, their bodies can recover quickly when they do drink water, rapidly rehydrating their tissues. Their kidneys are incredibly efficient, concentrating urine to minimize water loss. This, combined with their digestive efficiency and ability to derive water from food, minimizes water consumption. These strategies underscore camels' incredible adaptations to their environment, positioning them as iconic examples of desert survival.