Question

In one of your rigorous workout sessions, you lost \(150 . \mathrm{g}\) of water through evaporation. Assume that the amount of work done by your body was \(1.80 \cdot 10^{5} \mathrm{~J}\) and that the heat required to evaporate the water came from your body. a) Find the loss in internal energy of your body, assuming that the latent heat of vaporization is \(2.42 \cdot 10^{6} \mathrm{~J} / \mathrm{kg} .\) b) Determine the minimum number of food calories that must be consumed to replace the internal energy lost ( 1 food calorie \(=4186 \mathrm{~J}\) ).

Short answer

Answer: The loss in internal energy of the body is -1.83 x 10^5 J, and the minimum number of food calories needed to replace that lost energy is 43.7 food calories.

Step-by-step solution

Calculate the heat loss

To find the heat loss, we need to multiply the mass of water evaporated by the latent heat of vaporization. In this case, we are given a mass of \(150 \text{ g}\) and a latent heat of vaporization of \(2.42 \cdot 10^{6} \text{ J/kg}\). First, convert the mass from grams to kilograms: $$ 150 \text{ g} \times \frac{1\text{ kg}}{1000 \text{ g}} = 0.15 \text{ kg} $$ Now, calculate the heat loss: $$ q = m \times L $$ $$ q = 0.15\text{ kg} \times 2.42 \cdot 10^{6} \text{ J/kg} $$ $$ q = 3.63 \cdot 10^{5} \text{ J} $$

Calculate the change in internal energy

The change in internal energy \((\Delta U)\) can be found using the first law of thermodynamics: $$ \Delta U = q + W $$ In this case, the heat (\(q\)) is lost from the body, so it has a negative sign. The amount of work done (\(W\)) is given as \(1.80 \cdot 10^{5} \text{ J}\): $$ \Delta U = -3.63 \cdot 10^{5} \text{ J} + 1.80 \cdot 10^{5} \text{ J} $$ $$ \Delta U = -1.83 \cdot 10^{5} \text{ J} $$

Calculate the minimum number of food calories needed

To find the minimum number of food calories needed to replace the lost energy, we will first convert the energy loss in Joules to food calories. We're given the conversion factor of 1 food calorie = 4186 J: $$ \frac{-1.83 \cdot 10^{5} \text{ J}}{ 4186 \text{ J/calorie}} = -43.7 \text{ food calories} $$ Since we cannot consume negative calories, we report the absolute value of the result as the minimum number of food calories needed: $$ 43.7 \text{ food calories} $$ In conclusion, the loss in internal energy of the body is \(-1.83 \cdot 10^{5} \text{ J}\) and the minimum number of food calories needed to replace that lost energy is \(43.7\) food calories.

Key concepts

First Law of Thermodynamics

When we talk about the first law of thermodynamics, we're referring to a fundamental concept in physics that relates to energy conservation. Simply put, it states that energy cannot be created nor destroyed; it can only change form. In the realm of a workout session, where you're sweating out water, the first law plays a crucial role in understanding how your body manages energy.

The equation representing the first law is typically written as \[ \Delta U = Q - W \.\] Here, \( \Delta U \) is the change in internal energy of the system, \(Q\) represents the heat added to the system, and \(W\) is the work done by the system. In the case of the workout, the work done by your body to evaporate the water is energy leaving the system, and so is the heat used for the vaporization of sweat; both contribute to a decrease in your body's internal energy.

Understanding this heat-energy-work relationship helps to analyze energy needs and metabolism, which are crucial for maintaining good health and physical performance.

Internal Energy

Diving deeper into the notion of internal energy, this term encompasses the total energy contained within a system - in this case, your body. The internal energy of a human body includes the kinetic energy of all moving particles (like molecules and atoms) and the potential energy resulting from the forces between these particles. During exercise, when you lose water through evaporation, there's a shift in internal energy.

The fact that the internal energy decreases during this process can be a bit counterintuitive. When you sweat, heat energy is used to convert water on your skin from a liquid to a vapor, which is the principle behind cooling off. This heat loss manifests as a loss of internal energy because your body has transferred energy to the environment to achieve the phase change of the water (sweat).

Heat Transfer

The heat transfer process is central to the understanding of not only engineering and environmental science but also the human body's temperature regulation. Heat is a form of energy that can be transferred from one object to another, and it naturally flows from a warmer to a cooler object.

In the context of your workout, the human body transfers heat to the sweat on the skin surface. The latent heat of vaporization, a specific amount of heat required to convert a substance from liquid to vapor without changing its temperature, is the hidden player in the game. It's crucial for the evaporation process, which helps your body to cool down. The latent heat of vaporization value is quite significant for water, indicating that a considerable amount of heat is needed to vaporize sweat, thereby effectively cooling the body as the water molecules take away heat when they transition to a gaseous state.

Energy Conversion

Finally, energy conversion is another topic that seamlessly integrates into our workout scenario. It refers to the transformation of energy from one form to another. In biological systems, such as the human body, one of the most important conversions is that of chemical energy from food into kinetic energy for muscles and thermal energy for maintaining body temperature.

After an intense workout, the body's energy stores are depleted, not just from mechanical work but also from thermoregulation, which includes the act of sweating. To replenish these stores, one must consume food, which the body will convert back into internal energy. This cycle of conversion is continuous and a beautiful demonstration of the laws of thermodynamics at play in our daily lives. The efficiency of these energy conversions is vital for our endurance and overall health.