Question

Appendix \(\mathrm{B}\) lists the vapor pressure of water at various external pressures. (a) Plot the data in Appendix B, vaporpressure (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 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?

Short answer

(a) After plotting the data, the estimated vapor pressure of water at \(37^{\circ} \mathrm{C}\) body temperature is approximately 47 torr. (b) The data point at \(100^{\circ}\mathrm{C}\) and 760 torr is significant because it represents the normal boiling point of water at sea level and standard atmospheric conditions. (c) In a city at an altitude of 5000 ft above sea level with a barometric pressure of 633 torr, the boiling point of water is approximately \(92^{\circ}\mathrm{C}\). (d) In a city at an altitude of 500 ft below sea level with a barometric pressure of 774 torr, the boiling point of water is approximately \(102^{\circ}\mathrm{C}\).

Step-by-step solution

(a) Plot the data and estimate the vapor pressure

Plot the data in Appendix B, which provides vapor pressure (torr) versus temperature \(\left(^{\circ} \mathrm{C}\right)\) values. To estimate the vapor pressure of water at body temperature, \(37^{\circ} \mathrm{C}\), observe the plotted data and determine approximately where \(37^{\circ} \mathrm{C}\) lies on the x-axis (temperature). Then read the corresponding y-axis (vapor pressure) value at this point. This value is the estimated vapor pressure at body temperature.

(b) Explain the significance of the data point

The data point at \(100^{\circ} \mathrm{C}\) and 760 torr is significant because it represents the normal boiling point of water. At this temperature and vapor pressure (which is equal to 1 atmospheric pressure), water changes from liquid to vapor under standard atmospheric conditions. This is the temperature at which water will boil at sea level.

(c) Boiling point at 5000 ft above sea level

At an altitude of 5000 ft, the barometric pressure is 633 torr. To find out the temperature at which water would boil in a city at this altitude, locate the point on the graph where the vapor pressure is equal to the barometric pressure of 633 torr. Read the corresponding temperature on the x-axis, and this will give you the boiling point of water in this city.

(d) Boiling point at 500 ft below sea level

At an altitude of 500 ft below sea level, the barometric pressure is 774 torr. To find out the temperature at which water would boil in a city at this altitude, locate the point on the graph where the vapor pressure is equal to the barometric pressure of 774 torr. Read the corresponding temperature on the x-axis, and this will give you the boiling point of water in this city.

Key concepts

Boiling Point Elevation

Boiling point elevation is a phenomenon where the boiling point of a liquid increases when a non-volatile solute is dissolved in it. This happens due to the solute particles disrupting the solvent’s ability to evaporate, effectively requiring more energy (or a higher temperature) for the liquid to transition into its gaseous state.

Additionally, when considering water boiling at different elevations, boiling point elevation also refers to the change in boiling temperature due to air pressure variations. At higher altitudes, the atmospheric pressure is lower, which means that water will boil at a lower temperature compared to at sea level. Thus, a pot of water boils faster on a mountain top than at the beach, but because the boiling point is lower, it’s also less effective at cooking food. This can be of particular interest in processes like cooking, where adjustments for altitude may be necessary.

Atmospheric Pressure

Atmospheric pressure, also known as barometric pressure, is the force exerted by the weight of the earth's atmosphere on any surface. It decreases as altitude increases since there is less air above exerting a downward force.

At sea level, the standard atmospheric pressure is defined as 760 torr, which is equivalent to 1013.25 millibars or about 14.7 psi (pounds per square inch). Understanding atmospheric pressure is essential in various fields such as meteorology, aviation, and even cooking, because it affects weather patterns, flight altitudes, and the aforementioned boiling points of liquids. Changes in atmospheric pressure can signal shifting weather conditions and also influence how we regulate equipment like pressure cookers or determine the necessary adjustments for cooking times and temperatures at different elevations.

Phase Change

A phase change, also known as a phase transition, is a transformation in the state of matter, such as from solid to liquid or liquid to gas. These transitions occur when energy, usually in the form of heat, is added or removed from a substance.

For water, the transition from liquid to vapor is called evaporation or boiling, and it occurs when the vapor pressure of water equals the atmospheric pressure. At this point, the molecules have enough energy to break free from their intermolecular forces and escape into the air as gas. The phase change of water is not only an everyday occurrence, such as when boiling water for tea, but is also a fundamental concept in understanding weather patterns, engine cooling systems, and the Earth’s hydrological cycle.