Problem 120 - University Physics with Modern Physics

Question

Food Intake of a Hamster. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has density 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) and specific heat 1020 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . A 50.0 -g hamster is placed in a calorimeter that contains 0.0500 \(\mathrm{m}^{3}\) of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises 1.60 \(\mathrm{C}^{\circ}\) per hour. How much heat does the running hamster generate in an hour? Assume that all this heat goes into the air in the calorimeter. You can ignore the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings. (b) Assuming that the hamster converts seed into heat with an efficiency of 10\(\%\) and that hamster seed has a food energy value of 24 \(\mathrm{J} / \mathrm{g}\) , how many grams of seed must the hamster eat per hour to supply this energy?

Step-by-Step Solution

Answer

The hamster generates 97.92 J of heat per hour and needs to eat 40.8 grams of seed per hour to meet this energy requirement.

Step 1: Calculate the mass of the air in the calorimeter

Given the density of air is 1.20 kg/m³ and the volume of air in the calorimeter is 0.0500 m³. The mass of the air can be calculated using the formula:\[ \text{mass} = \text{density} \times \text{volume} \]Substituting the values:\[ \text{mass} = 1.20 \, \text{kg/m}^3 \times 0.0500 \, \text{m}^3 = 0.0600 \, \text{kg} \]

Step 2: Calculate the heat generated by the hamster

The formula for heat energy is:\[ Q = m \cdot c \cdot \Delta T \]where \( m \) is the mass of the air, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. Substituting the values:\[ Q = 0.0600 \, \text{kg} \times 1020 \, \text{J/kg} \cdot \text{K} \times 1.60 \, \text{K} \]\[ Q = 97.92 \, \text{J} \]Therefore, the hamster generates 97.92 J of heat per hour.

Step 3: Determine the amount of energy the hamster converts into food

The hamster has an efficiency of 10% in converting seed energy into heat energy. Thus, if \( Q \) is the heat generated, the food energy required is:\[ \text{Energy consumed} = \frac{Q}{\text{Efficiency}} = \frac{97.92 \, \text{J}}{0.10} \]\[ \text{Energy consumed} = 979.2 \, \text{J} \]

Step 4: Calculate the grams of seed needed based on energy value

Each gram of seed provides 24 J of energy. To find out how many grams are needed:\[ \text{mass of seed} = \frac{\text{Energy consumed}}{\text{Energy per gram}} = \frac{979.2 \, \text{J}}{24 \, \text{J/g}} \]\[ \text{mass of seed} = 40.8 \, \text{g} \]

Step 5: Conclusion

The hamster generates 97.92 J of heat per hour. To support this energy output, the hamster must consume approximately 40.8 grams of seed per hour.

Key Concepts

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes, and it provides vital insights into metabolic processes. In this exercise, a simple calorimeter is used to determine the Basal Metabolic Rate (BMR) of a hamster by measuring the heat generated while it runs in a wheel.
The calorimeter confines the hamster within an insulated box containing air, which acts as the medium for heat absorption. By observing the temperature increase of the air over time, one can deduce the amount of heat energy produced by the hamster.
Such measurements allow us to better understand energy transformations in biological systems. The calorimetry process in this context treats the air as a single system, assuming all heat generated is absorbed without loss. This simplification helps in focusing on the hamster's metabolic efficiency and energy requirements.

Heat Energy Calculation

Calculating the heat energy involves knowing the mass of the air inside the calorimeter, the specific heat capacity, and the change in temperature. The formula used is \[ Q = m \cdot c \cdot \Delta T \]where \( Q \) represents the heat energy, \( m \) is the mass of the air, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.

In this example, the air's density and the calorimeter's volume allow the computation of the air's mass, which combined with the given specific heat and temperature change, results in the total heat energy.

• The density of air provides a basis to calculate its mass: \(1.20 \, \text{kg/m}^3\).
• Given volume: \(0.0500 \, \text{m}^3\).
• Resulting mass: \(0.0600 \, \text{kg}\).

This energy calculation is pivotal to understanding how organisms like the hamster utilize food energy.

Metabolic Efficiency

Metabolic efficiency refers to how effectively an organism can convert energy from food into other forms of energy, such as heat. This exercise addresses how efficiently a hamster converts the energy from seeds it consumes into the heat energy it produces when running.

The efficiency percentage given (10%) implies that only a fraction of the food energy is utilized for generating heat, while the rest may be used in other biological processes or stored. This concept ensures that we take into consideration not just the calories consumed, but how these calories are utilized in the body.
Understanding metabolic efficiency is crucial for assessing dietary needs and energy balance in living organisms.

Specific Heat Capacity

Specific heat capacity is a key concept in calculating how much the temperature of a substance changes as it absorbs or loses heat. It is defined as the amount of heat per unit mass required to raise the temperature by one degree Celsius.

The air in the calorimeter has a specific heat capacity of \(1020 \, \text{J/kg} \cdot \text{K}\). This means that it requires 1020 joules of heat to raise the temperature of one kilogram of air by one degree Kelvin.

Understanding specific heat capacity is essential in determining how different substances react to heat energy, which is vital for processes like calorimetry where heat changes are measured. Specific heat capacity plays a crucial role in energy transformations and is integral to understanding thermal exchanges in biological and physical systems.