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 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 $=4186$ J).

Short answer

Answer: 44 food calories.

Step-by-step solution

Calculate the energy required to evaporate the given amount of water

To calculate the energy required to evaporate the given amount of water, we use the formula: Energy = Mass x Latent heat of vaporization where Mass = $150\text{g}$ (converted to kg) and Latent heat of vaporization = $2.42\cdot {10}^6\mathrm{J}/\mathrm{kg}$. Converting the mass of water to kg: $150\text{g}\times \frac{1\text{kg}}{1000\text{g}}=0.15\text{kg}$. Energy = $0.15\text{kg}\times 2.42\cdot {10}^6\mathrm{J}/\mathrm{kg}$ Energy = $3.63\cdot {10}^5\text{J}$

Determine the loss in internal energy

According to the conservation of energy, the work done by the body is equal to the energy required to evaporate the water plus the change in the internal energy of your body, Work done = Energy + Loss in internal energy Rearranging the equation, we get: Loss in internal energy = Work done - Energy Loss in internal energy = $1.80\cdot {10}^5\text{J}-3.63\cdot {10}^5\text{J}$ Loss in internal energy = $-1.83\cdot {10}^5\text{J}$ The negative sign indicates that the internal energy of your body has decreased. b) Determine the minimum number of food calories

Convert the Joules to food calories

To convert the lost energy from Joules to food calories, we use the conversion factor given, 1 food calorie = 4186 J Lost energy in food calories = $\frac{-1.83\cdot {10}^5\text{J}}{4186\mathrm{J}/\mathrm{food}\mathrm{calorie}}$ Lost energy in food calories = $-43.77\text{food calories}$

Find the minimum number of food calories needed to replace the lost internal energy

Since energy cannot be negative, we must consume at least 43.77 food calories to replace the lost internal energy. Minimum number of food calories = 44 (rounded up)

Key concepts

Internal Energy

Internal energy is a concept in thermodynamics that represents the total energy contained within a system due to both its thermal and molecular motion. It is a critical aspect when considering energy changes during processes like phase transitions or chemical reactions.

In the context of the exercise, the internal energy of your body changes as a result of evaporating sweat. When you sweat, the body uses its stored thermal energy to convert liquid sweat into vapor, which results in a loss of internal energy.

The solution outlines how the loss of internal energy is calculated. By applying the principle of energy conservation, we know:

• Energy needed to evaporate water + Work done by body = Change in internal energy.
• If the energy for evaporation exceeds the work done, internal energy decreases.

In our example, the internal energy loss calculated is negative, indicating an actual decrease in the body's internal energy, highlighting the energy expended in evaporating the sweat.

Latent Heat of Vaporization

Latent heat of vaporization is the amount of heat energy required to turn a substance from a liquid into a vapor without changing its temperature. It's an important concept in understanding phase changes in thermodynamics.

This heat is absorbed from the environment, or in the case of your body, from the internal energy stores, allowing the substance (sweat) to transition into gaseous form. In this specific exercise, we examine the latent heat of vaporization of water, quantified as $2.42\times {10}^6\mathrm{J}/\mathrm{kg}$.

Using the formula for energy needed in phase transition:
Energy = Mass $\times$ Latent heat of vaporization,
we can calculate the amount of energy required for the evaporation of $150\mathrm{g}$ of water. By converting grams to kilograms, and applying the latent heat value, the energy required is $3.63\times {10}^5\mathrm{J}$. Understanding this can help appreciate why sweating cools you down – it removes a substantial amount of heat from your body!

Energy Conservation

Energy conservation is a fundamental principle in physics, indicating that energy in a closed system remains constant. It implies that within such a system, energy can neither be created nor destroyed, only transformed from one form to another.

In the scenario of a workout, your body acts as this closed system. As you exercise and sweat, the energy transformation occurs in several stages:

• Internal energy is transformed to evaporate sweat (as discussed with internal energy).
• Work done by the body contributes to the energy balance equation.

The formula representing this is:
Work done = Energy for evaporation + Loss in internal energy. By rearranging this formula, it helps us understand how much internal energy the body loses, answering part of the exercise's question.

The exercise further emphasizes energy conservation by converting the lost energy into food calories. This step shows what is required to restore the internal energy balance. In doing so, it reinforces how energy is budgeted and used in our daily physiological processes.