Problem 3 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An average human produces about \(10 \mathrm{MJ}\) of heat each day through metabolic activity. If a human body were an isolated system of mass \(80 \mathrm{~kg}\) with the heat capacity of water, what temperature rise would the body experience? Heat capacity of water \(=4.2 \mathrm{~J} / \mathrm{K}-\mathrm{g} .\) (a) \(29.76^{\circ} \mathrm{C}\) (b) \(2.976 \mathrm{~K}\) (c) \(2.976 \times 10^{4 \circ} \mathrm{C}\) (d) \(0.029^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The temperature rise would be (a) 29.76 degrees Celsius.

Step 1: Understand the Concept of Heat Capacity

Heat capacity is the amount of heat required to raise the temperature of a given mass by one degree (Celsius or Kelvin). Here, we have the specific heat capacity of water, which is the heat capacity per unit mass.

Step 2: Calculate the Total Heat Capacity of the Body

Since the heat capacity of water is given per gram, we need to first convert the body's mass from kilograms to grams, which is 80 kg x 1000 g/kg = 80000 g. Then multiply the mass by the specific heat capacity of water to get the total heat capacity: Heat Capacity = Mass x Specific Heat Capacity = 80000 g x 4.2 J/K-g.

Step 3: Compute the Temperature Rise

To find the temperature rise, divide the amount of heat produced by the body (10 MJ, which is 10 x 10^6 J) by the total heat capacity. Temperature Rise = Heat Produced / Total Heat Capacity.

Step 4: Convert the Temperature Rise to Celsius or Kelvin

Since the specific heat capacity is given in J/K-g, and 1 degree Celsius is equal to 1 Kelvin, the temperature rise can be expressed directly in either Celsius or Kelvin without requiring a conversion.

Step 5: Solve for the Temperature Rise

Plugging in the values, we get Temperature Rise = (10 x 10^6 J) / (80000 g x 4.2 J/K-g) = 29.76 K, which is also 29.76 degrees Celsius since the increment is the same for both units.

Key Concepts

Metabolic Heat Production

Metabolic heat production is the energy released as heat by biochemical reactions in the body, particularly during the process of metabolism. This heat is essential in maintaining a constant internal body temperature despite varying environmental conditions.

When you undergo any physical activity or even while at rest, your cells are performing metabolism, which consists of breaking down nutrients to create energy. This energy is not just for moving or thinking, but also for the countless cellular activities that keep you alive. However, not all of this energy is utilized for work; much of it is lost as heat. This heat must be properly managed or it can lead to an increase in body temperature. Humans, and other mammals, have specialized means of regulating this heat through mechanisms like sweating and altering blood flow to the skin.

In the given exercise, the average human metabolic heat production is about 10 MJ (megajoules) per day. Without efficient heat dispersal, this would lead to an increase in body temperature, demonstrating the importance of thermoregulation.

Specific Heat Capacity

The specific heat capacity is a property of a material that indicates how much heat energy (in joules) is required to raise the temperature of one gram of the material by one degree Celsius (or Kelvin). The specific heat capacity of water, for example, is relatively high, meaning it takes a considerable amount of heat to raise its temperature.

This property is crucial in many biological processes, as organisms are mostly water-based. A high specific heat capacity of water allows for better temperature regulation within the body, preventing sudden changes in temperature, which could be detrimental to enzyme function and other biochemical reactions. In the context of the exercise, the heat capacity of the water is used to emulate the human body's heat capacity because our bodies are composed largely of water.

Temperature Change Calculation

Calculating temperature change involves determining how much an object's temperature will increase or decrease when a certain amount of heat is added or removed. This calculation requires knowledge of the specific heat capacity of the substance and the mass of the object being heated.

In the exercise, we applied the formula for temperature change, which is the heat produced divided by the product of the mass and the specific heat capacity. By carrying out this computation, the temperature change for the average human body based on metabolic heat production was found to be 29.76 degrees Celsius, indicating how much the body temperature would increase over one day if there were no means of heat dissipation.

Thermodynamics in Biochemistry

Thermodynamics plays a vital role in biochemistry, which deals with understanding the energy changes associated with biochemical reactions. The principles of thermodynamics help explain how organisms obtain and use energy, how energy is transformed within cells, and how it influences the larger biological systems.

In biochemistry, the body is often compared to a thermodynamic system. This comparison allows us to use concepts like specific heat capacity and metabolic heat production to analyze the thermal behaviors of living beings. The exercise highlights the application of thermodynamics by calculating the temperature change as a result of metabolic heat production using the concept of specific heat capacity, all rooted in the laws of thermodynamics that govern energy transformations in biochemical systems.

Understanding these concepts is key to appreciating how efficiently the human body operates and the intricate balance it maintains through the conversion and release of energy.