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Chapter 15: Problem 67

\(\bullet$$\bullet\) The effect of altitude on the lungs. (a) Calculate the change in air pressure you will experience if you climb a 1000 \(\mathrm{m}\) moun- tain, assuming that the temperature and air density do not change over this distance and that they were \(22^{\circ} \mathrm{Cand} 1.2 \mathrm{kg} / \mathrm{m}^{3},\) respectively, at the bottom of the mountain. (b) If you took a 0.50 \(\mathrm{L}\) breath at the foot of the mountain and managed to hold it until you reached the top, what would be the volume of this breath when you exhaled it there?

Short Answer

Expert verified
The pressure decreases by 11772 Pa, and the breath volume increases to 0.57 L.

Step by step solution

01

Understand the Problem

We are calculating the change in air pressure when climbing a 1000 m mountain, assuming no change in temperature and air density. Then we find the volume change of a 0.50 L breath when it is taken from the bottom to the top.
02

Calculate the Change in Air Pressure

Use the formula for pressure change with altitude: \( \Delta P = - \rho g \Delta h \), where \( \rho = 1.2 \, \text{kg/m}^3 \), \( g = 9.81 \, \text{m/s}^2 \), and \( \Delta h = 1000 \, \text{m} \). Substitute these values:\[ \Delta P = - (1.2 \, \text{kg/m}^3) (9.81 \, \text{m/s}^2) (1000 \, \text{m}) \]\[ \Delta P = - 11772 \, \text{Pa} \].
03

Calculate the Final Volume of Breath

Use Boyle's Law, \( P_1 V_1 = P_2 V_2 \), where \( P_1 \) is the initial pressure, \( V_1 = 0.50 \, \text{L} \), and \( V_2 \) is the final volume. Assume \( P_1 - \Delta P \) at the top.\Let \( P_1 \) be atmospheric pressure at sea level, approximately 101325 Pa:\[ 101325 \times 0.50 = (101325 - 11772) \times V_2 \]Solving for \( V_2 \):\[ 50662.5 = 89553V_2 \]\[ V_2 = \frac{50662.5}{89553} \approx 0.57 \, \text{L} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Altitude effect on pressure
As you climb higher altitudes, the air pressure decreases. This happens because the density of the air becomes less with increasing altitude since the atmosphere gets thinner. Gravity pulls air molecules towards the Earth, so the higher you go, the fewer air molecules are above you, leading to lower pressure.
  • At higher altitudes, there is less weight of the air above pushing down, which results in lower air pressure.
  • Since air pressure decreases, any given volume of air contains fewer molecules, affecting how we breathe and perceive climate.

Understanding the altitude effect on pressure is crucial, especially for activities like mountain climbing or flying, where altitude changes significantly.
Pressure change formula
The pressure change formula helps us calculate how pressure varies with altitude. For small vertical changes, we use: \[ \Delta P = - \rho g \Delta h \] Here, \( \Delta P \) represents the change in pressure, \( \rho \) is the air density, \( g \) is the gravitational acceleration (9.81 m/s虏), and \( \Delta h \) is the change in height.
  • The negative sign indicates a decrease in pressure as height increases.
  • Knowing the density and height change allows us to compute how much the pressure drops as one climbs upwards.

Using this formula allows us to understand and predict how pressure changes with elevation, which is particularly useful in meteorology, aerospace, and environmental science.
Lungs and altitude
Our lungs are directly affected by altitude changes due to the varying air pressure. As pressure decreases with higher altitude, the availability of oxygen also reduces, impacting breathing efficiency.
  • Lower air pressure means there is less oxygen in each breath, which can lead to altitude sickness if the body doesn't acclimatize.
  • Over time, the body may adapt by increasing red blood cell count to improve oxygen transport.

When ascending to high altitudes, it's important to understand these physiological changes and take necessary precautions. Activities like climbing or long stays at elevated areas demand acclimatization for health safety.
Air pressure at altitude
Air pressure at altitude is significantly different compared to sea level. At sea level, the atmospheric pressure is roughly 101325 Pa, whereas, at higher altitudes, it decreases because of reduced air density.
  • This change affects weather conditions and how items such as aircraft perform.
  • At 1000 meters, pressure drops significantly impacting both mechanical systems and living organisms.

Thus, calculating exact changes in pressure, like using Boyle's Law in exercises, is key for designing equipment and planning activities in different altitudes. Understanding these changes helps in various fields from engineering to health sciences.

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Most popular questions from this chapter

\(\bullet\) Two moles of an ideal gas are heated at constant pressure from \(T=27^{\circ} \mathrm{C}\) to \(T=107^{\circ} \mathrm{C}\) (a) Draw a \(p V\) diagram for this process. (b) Calculate the work done by the gas.

\(\bullet$$\bullet\) A flask with a volume of 1.50 \(\mathrm{L}\) , provided with a stopcock, contains ethane gas \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) at 300 \(\mathrm{K}\) and atmospheric pressure \(\left(1.013 \times 10^{5} \mathrm{Pa}\right) .\) The molar mass of ethane is 30.1 \(\mathrm{g} / \mathrm{mol}\) . The system is warmed to a temperature of \(380 \mathrm{K},\) with the stopcock open to the atmosphere. The stopcock is then closed, and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?

\(\bullet$$\bullet\) A flask contains a mixture of neon (Ne), krypton (Kr), and radon \((\mathrm{Rn})\) gases. Compare (a) the average kinetic energies of the three types of atoms; (b) their root-mean-square speeds. (Hint: The periodic table in Appendix C shows the molar mass (in \(\mathrm{g} / \mathrm{mol}\) ) of each element.)

\(\bullet\) Suppose some insects have speeds of \(10.00 \mathrm{m} / \mathrm{s}, 8.00 \mathrm{m} / \mathrm{s},\) \(7.00 \mathrm{m} / \mathrm{s},\) and 2.00 \(\mathrm{m} / \mathrm{s} .\) Find (a) the rms speed of these critters and (b) their average speed.

\(\bullet\) A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 \(\mathrm{m}^{3}\) of air at a pressure of 3.40 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 \(\mathrm{m}^{3} .\) If the temperature remains constant, what is the final value of the pressure?
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