Question

Appendix B lists the vapor pressure of water at various external pressures. (a) Plot the data in Appendix B, vapor pressure (torr) versus temperature \(\left({ }^{\circ} \mathrm{C}\right) .\) From your plot, estimate the vapor pressure of water at body temperature, \(37^{\circ} \mathrm{C}\). (b) Explain the significance of the data point at 760.0 torr, \(100^{\circ} \mathrm{C}\) (c) A city at an altitude of \(5000 \mathrm{ft}\) above sea level has a barometric pressure of 633 torr. To what temperature would you have to heat water to boil it in this city? (d) A city at an altitude of \(500 \mathrm{ft}\) below sea level would have a barometric pressure of 774 torr. To what temperature would you have to heat water to boil it in this city? (e) For the two cities in parts \((\mathrm{c})\) and \((\mathrm{d}),\) compare the average kinetic energies of the water molecules at their boiling points. Are the kinetic energies the same or different? Explain.

Short answer

(a) After plotting the data from Appendix B, we can estimate the vapor pressure of water at body temperature ($37^{\circ} \mathrm{C}$) to be approximately 47 torr. (b) The data point at 760.0 torr and $100^{\circ} \mathrm{C}$ is significant because it represents the boiling point of water at standard atmospheric pressure (1 atm), where water changes from its liquid phase to its gas phase. (c) In the city at an altitude of $5000 \mathrm{ft}$ with a barometric pressure of 633 torr, you would have to heat water to approximately $94^{\circ} \mathrm{C}$ to boil it. (d) In the city at an altitude of $500 \mathrm{ft}$ below sea level with a barometric pressure of 774 torr, you would have to heat water to approximately $102^{\circ} \mathrm{C}$ to boil it. (e) The average kinetic energies of the water molecules at boiling points in the two cities are the same, as the kinetic energy needed to change water from liquid to gas (boiling point) remains constant regardless of external pressure.

Step-by-step solution

Plot the Data

Use the data provided in Appendix B to create a plot of vapor pressure (in torr) versus temperature (in °C). You can use graph paper or a computer program like Excel to draw the graph. Make sure to label the axes correctly.

Estimate Vapor Pressure at 37°C

Look for the point on the graph that corresponds to 37°C (body temperature) on the x-axis (temperature). Then, draw a horizontal line from that point to the curve, and from the curve to the y-axis (vapor pressure) to read the approximate vapor pressure at 37°C. (b)

Significance of the Data Point

The data point at 760.0 torr and 100°C is significant because it represents the boiling point of water at standard atmospheric pressure (1 atm, equivalent to 760 torr). At this point, water will change from its liquid phase to its gas phase. (c)

Determine Boiling Point at High Altitude

At an altitude of 5000 ft, with a barometric pressure of 633 torr, find the temperature on the x-axis (temperature) corresponding to a vapor pressure of 633 torr on the y-axis. Draw a horizontal line from 633 torr to the curve and a vertical line from the curve to the x-axis to read the temperature at which water boils in this city. (d)

Determine Boiling Point Below Sea Level

At an altitude of 500 ft below sea level, with a barometric pressure of 774 torr, find the temperature on the x-axis (temperature) corresponding to a vapor pressure of 774 torr on the y-axis. Draw a horizontal line from 774 torr to the curve and a vertical line from the curve to the x-axis to read the temperature at which water boils in this city. (e)

Compare Kinetic Energies of Water Molecules

The average kinetic energies of the water molecules at boiling points in the two cities are the same. This is because, regardless of external pressure, the kinetic energy needed to change water from its liquid phase to its gas phase (boiling point) remains constant. At the boiling point, the boiling water has enough kinetic energy to overcome the intermolecular forces, causing the phase change.

Key concepts

Boiling Point

The boiling point of a substance is the temperature at which its vapor pressure equals the external pressure surrounding it. For water, this typically occurs at 100°C under standard atmospheric pressure (1 atm or 760 torr). At this point, water undergoes a phase change from liquid to gas as it boils.

The boiling point can change based on environmental conditions, particularly pressure. If the external pressure decreases, such as at higher altitudes, the boiling point of water decreases. Conversely, it increases when the external pressure, such as below sea level, is higher than standard pressure. Therefore, the boiling point is not a constant value but varies with atmospheric conditions.

Kinetic Energy

Kinetic energy relates to the energy possessed by molecules due to their motion. At the boiling point, the average kinetic energy of water molecules is sufficient to break intermolecular forces, allowing the phase transition from liquid to gas. Despite differing boiling temperatures due to pressure variations, the kinetic energy needed for water molecules to overcome these forces remains unchanged.

That means whether you are high on a mountain or below sea level, once water reaches its boiling point, the average kinetic energy of its molecules is uniform. The temperature might vary, but the energy barrier they need to overcome is consistent, which explains why boiling occurs.

Altitude Effects

Elevation above or below sea level significantly impacts water's boiling point due to changes in atmospheric pressure. At higher altitudes, air pressure is lower, which results in a decreased boiling point. For instance, in a city 5000 ft above sea level, water boils at a temperature lower than 100°C because the external pressure is around 633 torr.

This decrease in boiling point can affect cooking and other processes that depend on water boiling. At altitudes below sea level where atmospheric pressure is higher, like a city 500 ft below sea level, water boils above 100°C since the pressure exceeds 760 torr. Adjustments might be needed to accommodate these changes in boiling temperatures due to altitude effects.

Water Phase Change

When water heats to its boiling point, it undergoes a phase change from liquid to gas. This transformation happens when water molecules gain enough kinetic energy to escape from intermolecular attractions. This stage is marked by a rapid rise in vapor pressure matching the atmospheric pressure.

During a phase change, the temperature remains constant even though heat energy continues to be supplied. Instead, this energy is used to break the bonds holding water in a liquid form, allowing molecules to transition to the gas phase. Understanding this phase change is essential in explaining why certain temperatures are needed for boiling under various atmospheric pressures.