Problem 81 - College Physics

Question

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 a density of \(1.29 \mathrm{~kg} / \mathrm{m}^{3}\) and a specific heat of \(1020 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K}) . \mathrm{A} 50.0 \mathrm{~g}\) 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 \(1.8 \mathrm{C}^{\circ}\) 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. 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 the energy found in part (a)?

Step-by-Step Solution

Answer

The hamster generates about 118.67 J of heat and needs about 49.45 grams of seed per hour.

Step 1: Calculate the Heat Added to the Air

To find out how much heat the hamster generates, we first need to use the formula for heat energy:\[ Q = mc\Delta T \]Here, \( m \) is the mass of air, \( c \) is the specific heat capacity of the air (\(1020 \text{ J/(kg}\cdot \text{K)}\)), and \( \Delta T \) is the change in temperature. The mass of air can be determined from its density and volume:\[ m = \rho \times V \]\( \rho = 1.29 \text{ kg/m}^3 \) and \( V = 0.0500 \text{ m}^3 \) gives:\[ m = 1.29 \times 0.0500 = 0.0645 \text{ kg} \]Thus, substituting these values, we have:\[ Q = 0.0645 \times 1020 \times 1.8 = 118.674 \text{ J/hr} \]The heat generated by the hamster in one hour is approximately \( 118.67 \text{ J} \).

Step 2: Calculate the Total Energy Requirement

Since only 10% of the seed energy is converted into heat, we consider the total energy required by the hamster. If the hamster generates \(118.67 \text{ J} \) of heat, this is 10% of the energy provided by the seed:\[ \text{Energy input from food} = \frac{Q}{\text{efficiency}} = \frac{118.67}{0.10} = 1186.7 \text{ J} \]

Step 3: Determine the Mass of Seed Required

The seed has a food energy value of \(24 \text{ J/g} \). Thus, to provide \(1186.7 \text{ J} \) of energy, the hamster needs:\[ \text{mass of seed} = \frac{1186.7}{24} = 49.446 \text{ g} \]The hamster needs approximately \( 49.45 \text{ g} \) of seed per hour.

Key Concepts

Calorimetry

Calorimetry is a fascinating branch of science aimed at measuring the amount of heat exchanged in chemical reactions or physical changes. In biological contexts, such as with our energetic hamster, it helps in tracking how much heat an organism generates.
To measure heat output, a calorimeter—a well-insulated container—is used. It ensures minimal heat transfer with the environment, thereby giving accurate readings of heat generated by the subject, based on temperature changes within.
In the exercise, the calorimeter enables the measurement of the evolved heat from the hamster as it runs, demonstrating how it expends energy to perform tasks.

Energy Conversion

Energy conversion plays a crucial role in understanding how living organisms function. When a hamster consumes food, it converts the energy stored in that food into mechanical energy and heat.
This exercise demonstrates an energy conversion process where the hamster's activity (running on the wheel) is powered by the energy derived from seed. While specific aspects such as the hamster's muscle function aren't directly observable in this scenario, the calorimeter provides insights into the amount of energy converted to heat.
Such conversions involve complicated biological processes that ensure that organisms make efficient use of their food intake.

Heat Energy Calculation

Heat energy calculation is an essential tool in calorimetry, allowing us to quantify the energy released or absorbed in a system. By utilizing the formula:\[ Q = mc\Delta T \]
where \( Q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change, we can understand how much energy has been transformed as a result of an organism's metabolic activities.
In the exercise, this calculation gives us an insight into how much energy the hamster uses while running, which is reported as the heat transferred to the surrounding air.

Specific Heat Capacity

Specific heat capacity is a material's ability to absorb heat per unit mass per degree temperature change. It's a measure of how much energy a substance can store.
The exercise uses the specific heat capacity of air, which is given as \(1020\text{ J/(kg} \cdot \text{K)}\). This value indicates how much energy is necessary to raise the temperature of 1 kilogram of air by one degree Celsius.
This parameter is crucial in determining how much heat the air around the hamster can absorb, helping us find out how much energy the hamster releases in the calorimeter.

Metabolic Efficiency

Metabolic efficiency explains the proportion of input energy (food energy) that an organism can convert into usable work or heat. In our example, the hamster's metabolic efficiency is 10%.
This means only 10% of the hamster's dietary energy is converted to heat energy, with the rest potentially being used for other biological functions such as growth or repair, or simply lost as inefficiency in metabolism.
The calculation of metabolic efficiency helps in understanding how much food a hamster must consume to match its energy output, making it crucial for calculating dietary needs in relation to physical activity.