Question

It has been determined that the body can generate $5500\mathrm{kJ}$ of energy during one hour of strenuous exercise. Perspiration is the body's mechanism for eliminating this heat. How many grams and how many liters of water would have to be evaporated through perspiration to rid the body of the heat generated during two hours of exercise? (The heat of vaporization of water is $40.6\mathrm{kJ}/\mathrm{mol}.)$

Short answer

Total heat generated during two hours of exercise is $2\times 5500\mathrm{kJ}=11000\mathrm{kJ}$. To find the moles of water needed to be evaporated, we can use the heat of vaporization: Moles of water = $\frac{11000\mathrm{kJ}}{40.6\mathrm{kJ}/\mathrm{mol}}\approx 271.0\mathrm{mol}$. Now, to find the grams of water: Grams of water = $271.0\mathrm{mol}\times 18.015\mathrm{g}/\mathrm{mol}\approx 4878.1\mathrm{g}$. And finally, to find the liters of water: Liters of water = $\frac{4878.1\mathrm{g}}{1000\mathrm{g}/\mathrm{L}}\approx 4.88\mathrm{L}$. Therefore, approximately 4878 grams (or 4.88 liters) of water would have to be evaporated through perspiration to rid the body of the heat generated during two hours of exercise.

Step-by-step solution

Calculate the total heat generated in two hours of exercise

To determine the amount of heat that the body generates during two hours of exercise, we multiply the heat generated per hour by the number of hours: Total heat (in kJ) = $5500\mathrm{kJ}/\mathrm{hour}\times 2\mathrm{hours}$

Find the moles of water needed to be evaporated

Now, we will use the heat of vaporization of water to convert the heat generated from two hours of exercise to moles of water: Moles of water = $\frac{\text{Total heat}}{\text{Heat of vaporization}}$

Calculate the grams of water

To determine the grams of water, we will multiply the moles of water by the molar mass of water (which is $18.015\mathrm{g}/\mathrm{mol}$): Grams of water = Moles of water × Molar mass of water

Calculate the liters of water

The last step is to convert the grams of water to liters using the density of water (which is $1\mathrm{g}/\mathrm{mL}$ or $1000\mathrm{g}/\mathrm{L}$): Liters of water = $\frac{\text{Grams of water}}{\text{Density of water}}$

Key concepts

perspiration

Perspiration is a crucial process by which our body regulates temperature. During activities like strenuous exercise, the body generates a significant amount of heat, and this heat needs to be eliminated to prevent overheating. The skin releases sweat, which is composed mostly of water. When this sweat evaporates, it absorbs heat from the body, effectively cooling us down. This process requires energy, known as the heat of vaporization. The body uses this mechanism to transfer the excess heat into the surroundings, maintaining a stable internal temperature. Without perspiration, physical activities could lead to dangerous increases in body temperature.

moles of water

To understand how much perspiration is required to eliminate body heat, it's essential to calculate the number of moles of water needed. The amount of heat absorbed when water turns from liquid to vapor is given by its heat of vaporization. For instance, if the body generates 11000 kJ of heat (from 2 hours of exercise), the moles of water can be calculated as follows:$$\text{Moles of water}=\frac{11000\text{kJ}}{40.6\text{kJ/mol}}$$This formula shows the relationship between the heat energy and the moles of water. Each mole of water absorbs 40.6 kJ of energy upon evaporation, making it an efficient means to dissipate body heat.

grams to liters conversion

Once the moles of water necessary for cooling are determined, we can convert this measurement into grams. This step involves using the molar mass of water:$$\text{Grams of water}=\text{Moles of water}\times 18.015\text{g/mol}$$Afterwards, to express this quantity in liters, we apply the concept of density. The density of water is typically 1 g/mL, which equals 1000 g/L. The conversion from grams to liters makes it easier to visualize the physical amount of water needed to be evaporated. The formula to convert grams into liters is:$$\text{Liters of water}=\frac{\text{Grams of water}}{1000\text{g/L}}$$

density of water

Density is a property of matter that helps determine how much space a certain mass will occupy. For water, the density is 1 g/mL, which makes conversions straightforward, as 1 gram of water is equivalent to 1 milliliter. This conversion is vital when determining how much water in liters needs to evaporate for the cooling process facilitated by perspiration: - **Simple Conversion**: Grams divided by the density of water (1000 g/L) convert directly to liters. - **Predictable Density**: Water's consistent density under typical conditions means calculations remain reliable. This property ensures that the calculations for the volume of evaporated water are accurate, crucial for understanding the heat dissipation needs during sweat evaporation.