Problem 86 - College Physics

Question

. Basal metabolic rate. 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 capacity of 1020 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) .\) 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. Neglect 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

(a) 118.404 J of heat is generated. (b) 49.34 g of seed is required.

Step 1: Determine the Mass of Air in the Calorimeter

The air volume in the calorimeter is given as 0.0500 m³. Using the density of air, which is 1.29 kg/m³, we can find the mass of air using the formula:\[ \text{mass of air} = \text{density of air} \times \text{volume of air} \]\[ \text{mass of air} = 1.29 \ \text{kg/m}^3 \times 0.0500 \ \text{m}^3 \]\[ \text{mass of air} = 0.0645 \ \text{kg} \]

Step 2: Calculate the Heat Absorbed by the Air

The specific heat capacity of air is 1020 J/(kg·K) and the temperature change is 1.8°C (or 1.8 K since °C to K changes are equivalent in magnitude). The heat absorbed by the air can be calculated using the formula:\[ Q = mc\Delta T \]Where:- \( m \) is the mass of the air (0.0645 kg)- \( c \) is the specific heat capacity (1020 J/(kg·K))- \( \Delta T \) is the temperature change (1.8 K)\[ Q = 0.0645 \ \text{kg} \times 1020 \ \text{J/(kg·K)} \times 1.8 \ \text{K} \]\[ Q = 118.404 \ \text{J} \]Thus, the heat generated by the hamster in an hour is 118.404 J.

Step 3: Adjust for Hamster's Efficiency in Converting Seed to Heat

Hamsters convert food energy into heat with an efficiency of 10%. This means that only 10% of the energy from the food is converted into heat. Therefore, the total energy intake from food should be 10 times the heat energy calculated:\[ \text{Total energy intake} = \frac{Q}{0.10} \]\[ \text{Total energy intake} = \frac{118.404 \ \text{J}}{0.10} \]\[ \text{Total energy intake} = 1184.04 \ \text{J} \]

Step 4: Calculate the Required Amount of Seed

The food energy value of hamster seed is 24 J/g. To find out how many grams of seed the hamster must eat to generate the energy found in Step 3, use the formula:\[ \text{mass of seed} = \frac{\text{Total energy intake}}{\text{energy per gram of seed}} \]\[ \text{mass of seed} = \frac{1184.04 \ \text{J}}{24 \ \text{J/g}} \]\[ \text{mass of seed} \approx 49.34 \ \text{g} \]

Step 5: Conclusion

(a) The running hamster generates approximately 118.404 J of heat in an hour. (b) It needs to eat approximately 49.34 grams of seed per hour to supply the energy required.

Key Concepts

Calorimetry

Calorimetry is a fascinating field in physics that deals with measuring the amount of heat exchanged during physical reactions or state changes. In the context of the basal metabolic rate (BMR), calorimetry helps measure how much energy, in the form of heat, is produced by the metabolism of an animal, such as a hamster. By using a calorimeter, a device designed for this purpose, we can track the temperature change of the air within it as the hamster runs.

The calorimeter works by providing an isolated environment where all heat produced is absorbed by the air inside, allowing for precise measurement. This involves using the relationship between heat absorbed, mass of air, specific heat capacity, and temperature change. By applying the formula \( Q = mc\Delta T \), we can calculate the total heat produced when knowing these variables.

This tells us not just about the hamster's activity but also about its energy expenditure through the conversion of food into heat, reflecting its metabolic rate.

Specific Heat Capacity

Specific heat capacity is an essential concept in thermal physics and calorimetry. It defines how much heat is required to change the temperature of a unit mass of a substance by one degree. In simpler terms, it's a measure of how well a substance holds onto heat.

For air, the specific heat capacity is around 1020 J/(kg·K). This means for each kilogram of air, 1020 joules are needed to raise its temperature by one Kelvin. This property allows us to calculate the energy exchange that happens in a calorimeter like the one used for measuring the hamster's metabolic rate.

Using the specific heat capacity, along with the known mass of air and the change in temperature, we can accurately determine the heat produced by metabolic processes. In our example, the heat calculated (118.404 J) reflects the hamster's basal metabolic activity, highlighting the application of specific heat capacity in practical exercises.

Energy Conversion

Energy conversion is a critical aspect of understanding metabolic processes. For animals, it involves converting the chemical energy from food into other forms of energy, such as heat during physical activity.

In the hamster's case, we consider how efficiently it converts food energy into thermal energy, which is necessary to calculate how much food it must consume to sustain its activity. Since only a portion of the energy consumed is converted into heat, knowing the efficiency rate is crucial.

With an efficiency of 10%, only one-tenth of the energy from the seed the hamster eats becomes heat. Therefore, to match the energy expenditure calculated for running, the hamster needs to consume ten times more energy from food, emphasizing the vital role of energy conversion efficiency in calculating metabolic needs.

Thermal Physics

Thermal physics is the branch of physics that deals with understanding the energy transfer processes resulting from temperature differences. It encompasses concepts such as heat, temperature, and energy flow.

In the hamster experiment, thermal physics principles are at play as we analyze how the running hamster transforms chemical energy from food into thermal energy. This study helps us not only understand energy conversion but also how living organisms maintain their energy balance.

By applying thermal physics, we can systematically analyze and quantify heat production, interactions between matter (such as air and the hamster in this case), and the resulting temperature changes. Such concepts form the backbone of understanding not only biological processes but also various physical phenomena in our daily lives.