Problem 22 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

If at top of a hill \(2000 \mathrm{~m}\) above sea level, the atmospheric pressure is \(50 \mathrm{~cm}\) of \(\mathrm{Hg}\) and at the sea level the atmospheric pressure is \(74.5 \mathrm{~cm}\) of \(\mathrm{Hg}\), and you need as much oxygen to breath at sea level as on the top of the hill, how much faster need you breathe on the hill top? (a) \(2.44\) times (b) \(1.49\) times (c) 5 times (d) 7 times

Step-by-Step Solution

Answer

You need to breathe 1.49 times faster on the hill top.

Step 1: Identify the Pressure Difference

Determine the atmospheric pressure at the top of the hill and at sea level. At the top of the hill, the pressure is 50 cm of Hg, and at sea level, it is 74.5 cm of Hg.

Step 2: Calculate the Ratio of Pressures

To find out how much faster you need to breathe on the hill top, calculate the ratio of sea-level pressure to hill top pressure: ratio = \( \frac{{pressure_{sea level}}}{{pressure_{hill top}}} = \frac{74.5}{50} \).

Step 3: Determine the Breathing Rate Factor

Divide the sea level pressure by the hill-top pressure to find out the factor by which you need to breathe faster on the hill to intake the same amount of oxygen: factor = \( \frac{74.5}{50} = 1.49 \).

Key Concepts

Physical Chemistry

Physical chemistry is a branch of chemistry concerned with the study of the properties and changes of matter, as well as the energy and dynamics that accompany such transformations. It investigates how chemical structure impacts the physical properties of substances and combines principles from physics and chemistry to understand the chemical behavior of matter at both the macroscopic and microscopic levels.

In the context of our exercise, physical chemistry helps us understand how changes in atmospheric pressure, which is a physical property, can influence the chemical behavior of gases, such as the rate at which oxygen is consumed by the human body during breathing. This field also considers how temperature, volume, and pressure—variables central to the gas laws—affect gases' behavior.

Gas Laws

Gas laws are a set of rules that describe the behavior of gases based on temperature, pressure, and volume. The most fundamental laws are Boyle's Law, Charles's Law, Gay-Lussac's Law, Avogadro's Law, and the Ideal Gas Law. These laws help us predict how changes in one property can affect another to maintain equilibrium.

For the purpose of our exercise, Boyle's Law is particularly relevant, which states that the pressure of a gas tends to decrease as the volume it occupies increases, provided the temperature remains constant. Thus, as we move higher above sea level, the atmospheric pressure decreases, and the volume of air (thus the amount of oxygen) one can intake in a single breath also changes, which is why breathing rate must be adjusted.

Oxygen Intake

Oxygen intake refers to the process of inhaling air containing oxygen into the lungs and transferring it into the bloodstream. It plays a critical role in respiration, a process essential for life, as all the cells in the body need oxygen to function correctly. The amount of oxygen absorbed depends on several factors, including the concentration of oxygen in the air, the breathing rate, and lung capacity.

In higher altitudes, as per our exercise, because the atmospheric pressure is lower, the concentration of oxygen is reduced. To compensate for the decreased availability of oxygen, one must breathe more rapidly to satisfy the body's demands. This is reflected in the exercise's requirement to calculate the increased breathing rate needed to intake the same amount of oxygen at the top of the hill as at sea level.

Pressure-Altitude Relationship

The pressure-altitude relationship is an understanding of how atmospheric pressure changes with altitude. As altitude increases, the atmospheric pressure decreases geometrically. This is because the density of air decreases with altitude; there are fewer air molecules to exert pressure. The relationship is predictable and is characterized by a roughly 11.3 pascal decrease in pressure for every meter increase in altitude.

In the given exercise, this concept allows us to understand why a person must breathe faster at the top of a hill, which is 2000 meters above sea level, than at sea level. The atmospheric pressure at the top of the hill is only 50 cm of Hg as compared to 74.5 cm of Hg at sea level; therefore, the amount of oxygen per breath taken at the hilltop will be lower, necessitating an increased breathing rate to achieve the same oxygen intake.