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 \((\mathrm{BMR})\) 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 \(1020 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K} . \mathrm{A} 50.0-\mathrm{g}\) 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 98.0 J of heat per hour and needs to consume 41 g of seed to meet its energy requirement.

Step 1: Calculate the Mass of the Air

The density of the air is given as \(1.20 \, \mathrm{kg/m^3}\), and the volume of the air is \(0.0500 \, \mathrm{m^3}\). To find the mass of air, use the formula \( \text{mass} = \text{density} \times \text{volume}\). Thus, the mass of the air is \(1.20 \, \mathrm{kg/m^3} \times 0.0500 \, \mathrm{m^3} = 0.060 \, \mathrm{kg}\).

Step 2: Determine Heat generated in Air

With the air's mass calculated, use the specific heat formula \(Q = mc\Delta T\) to find the heat \(Q\). Here, \(c = 1020 \, \mathrm{J/kg \, K}, \, m = 0.060 \, \mathrm{kg}, \, \Delta T = 1.60 \, \mathrm{C^\circ}\). Thus, \(Q = 0.060 \, \mathrm{kg} \times 1020 \, \mathrm{J/kg \, K} \times 1.60 \, \mathrm{K} = 98.0 \, \mathrm{J}\).

Step 3: Calculate Energy Requirement from Seed

The hamster converts seed into energy with only 10\(\%\) efficiency. Thus, to generate 98.0 J of heat, the hamster needs \(98.0 \, \mathrm{J} / 0.10 = 980 \, \mathrm{J}\) from the seed.

Step 4: Convert Energy Requirement to Seed Mass

Knowing that each gram of seed provides 24 J, calculate the mass of seed necessary: \( \text{mass of seed} = \text{required energy} / \text{energy per gram} \). Thus, \(980 \, \mathrm{J} / 24 \, \mathrm{J/g} = 40.83 \, \mathrm{g}\). Round to a practical value: about 41 g.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property of a substance that tells us how much heat energy is required to change the temperature of a given mass of the substance by one degree Celsius. It is typically represented by the symbol \( c \) and measured in joules per kilogram per degree Celsius (\( \text{J/kg·°C} \)).

In our exercise, the air inside the calorimeter has a specific heat capacity of 1020 \( \text{J/kg·K} \). This means that for each kilogram of air, 1020 joules are needed to increase the temperature by 1 Kelvin (which is numerically equal to 1°C in the context of this exercise).

Utilizing the formula \( Q = mc\Delta T \), where \( Q \) is the heat added, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change, we can determine the amount of heat absorbed by the air. By doing this calculation, we determined that the hamster generates 98.0 Joules of energy per hour.

Efficiency of Energy Conversion

Efficiency is the measure of how much useful energy is obtained from a given amount of energy input. In the context of our calorimetry experiment, efficiency tells us how well the hamster can convert the energy from the food (seeds) it eats into heat energy.

In the given scenario, the hamster's conversion efficiency is only 10\(\%\). This means that for every 100 joules of energy consumed, only 10 joules are converted into heat energy, while the rest is lost or used in other bodily functions.

To find out how much energy the hamster must consume to generate 98.0 Joules of heat, we need to consider its efficiency. By using the equation \( \text{Required Energy} = \frac{\text{Heat Generated}}{\text{Efficiency}} \), we can calculate that the hamster must take in 980 Joules of energy from its food to meet its metabolic needs.

Calorimetry

Calorimetry is a method used to measure the amount of heat released or absorbed by a substance. It is an essential technique in chemistry and biology for understanding metabolic rates and energy exchanges.

In our exercise, a calorimeter is used to determine the hamster's basal metabolic rate (BMR) by measuring the temperature increase of the air. The calorimeter is designed to be an isolated system in which the heat generated by the hamster is absorbed only by the air, allowing accurate measurement of the hamster's heat production without external heat losses.

The precision of calorimetry makes it a powerful tool to study the metabolism of different organisms. By providing insights into the heat produced by biological processes, calorimetry helps us understand how efficiently animals, like our hamster, use their food to sustain their activities.

Hamster Metabolism

Hamsters, like all living organisms, require energy to perform various functions, such as maintaining body temperature, running, and digesting food. This energy comes from the food they consume, which must be converted into usable forms through metabolic processes.

The basal metabolic rate (BMR) is a measurement of the minimum amount of energy expended at rest to maintain basic physiological functions. For the hamster in our exercise, its metabolic activity while running in a wheel raises the air temperature, indicating its active energy conversion. This higher energy expenditure is why the hamster needs to replenish its energy by eating seeds.

Understanding hamster metabolism, especially BMR, is vital for ensuring their dietary needs are met efficiently. By converting food energy with known efficiency—a concept reflected in our exercise—we can determine exactly how much food a hamster needs to maintain energy balance while being active.