Question

Appendix B lists the vapor pressure of water at various external pressures. (a) Plot the data in Appendix \(B\), vapor pressure (torr) vs. 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 (c) and (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

The vapor pressure of water at body temperature \(37^{\circ}\mathrm{C}\) can be estimated from a vapor pressure vs. temperature plot using the data in Appendix B. The normal boiling point of water is significant due to being at a standard atmospheric pressure of 760.0 torr and temperature of \(100^{\circ}\mathrm{C}\). For a city at an altitude of 5000 ft with a barometric pressure of 633 torr and a city at 500 ft below sea level with a barometric pressure of 774 torr, the boiling points of water can also be found using the vapor pressure vs. temperature plot. Comparing the average kinetic energies of water molecules at their boiling points in these cities reveals that they will be different if the boiling point temperatures in Kelvin are different, as average kinetic energy is proportional to temperature.

Step-by-step solution

Plot the vapor pressure data

The first task is to plot the vapor pressure (torr) vs. temperature (°C) using the data in Appendix B. Create a graph with temperature on the x-axis and vapor pressure on the y-axis.

Estimate the vapor pressure at body temperature

Once the plot is created in Step 1, locate \(37^{\circ}\mathrm{C}\) on the x-axis and estimate the corresponding vapor pressure on the y-axis. This value is the vapor pressure of water at body temperature.

Explain the significance of the data point at 760.0 torr, 100°C

The data point at \(760.0\) torr and \(100^{\circ}\mathrm{C}\) represents the normal boiling point of water. This is the temperature and pressure at which water transitions from the liquid to the vapor phase under standard atmospheric pressure (1 atm, or 760 torr).

Calculate the boiling point of water in a city at 5000 ft above sea level

For a city at an altitude of \(5000\) ft with a barometric pressure of 633 torr, use the vapor pressure vs. temperature plot. Locate the 633 torr value on the y-axis and find the corresponding temperature on the x-axis. This temperature is the boiling point of water in this city.

Calculate the boiling point of water in a city at 500 ft below sea level

For a city at an altitude of \(500\) ft below sea level with a barometric pressure of 774 torr, use the vapor pressure vs. temperature plot. Locate the 774 torr value on the y-axis and find the corresponding temperature on the x-axis. This temperature is the boiling point of water in this city.

Compare average kinetic energies

The average kinetic energy of water molecules at their boiling points in the two cities in parts (c) and (d) can be compared using the relationship between temperature and kinetic energy. The average kinetic energy of molecules is proportional to temperature (in Kelvin), i.e., \(KE_{avg} \propto T\). Convert the boiling point temperatures found in Steps 4 and 5 to Kelvin (adding 273.15 to the values) and compare them. If the temperatures are the same, the average kinetic energies are the same. If the temperatures are different, the average kinetic energies are different. Explain your findings based on the relationship between temperature and kinetic energy.