Question

The average lung capacity of a human is 6.0 L. How many moles of air are in your lungs when you are in the following situations? a. At sea level \((T=298 \mathrm{K}, P=1.00 \mathrm{atm})\) b. \(10 . \mathrm{m}\) below water \((T=298 \mathrm{K}, P=1.97 \mathrm{atm})\) c. At the top of Mount Everest \((T=200 . \mathrm{K}, P=0.296 \mathrm{atm})\)

Short answer

In the three given situations, the number of moles of air in the lungs can be calculated as follows: a. At sea level (T=298 K, P=1.00 atm) - \(n = 0.246 \mathrm{mol}\) b. 10 m below water (T=298 K, P=1.97 atm) - \(n = 0.483 \mathrm{mol}\) c. At the top of Mount Everest (T=200 K, P=0.296 atm) - \(n = 0.108 \mathrm{mol}\)

Step-by-step solution

Write down the given variables

For this situation, we have: - P = 1.00 atm - V = 6.0 L - T = 298 K - R = 0.0821 Latm / Kmol (Ideal Gas Constant)

Calculate the number of moles (n) using the Ideal Gas Law

Rearrange the Ideal Gas Law formula to solve for n: \(n = \frac{PV}{RT}\) Plug in the given values and calculate n: \(n = \frac{(1.00 \mathrm{atm})(6.0 \mathrm{L})}{(0.0821 \frac{\mathrm{Latm}}{\mathrm{Kmol}}) (298 \mathrm{K})} = 0.246 \mathrm{mol}\) So, at sea level, there are 0.246 moles of air in the lungs. #b. 10 m below water (T=298 K, P=1.97 atm)#

Write down the given variables

For this situation, we have: - P = 1.97 atm - V = 6.0 L - T = 298 K - R = 0.0821 Latm / Kmol (Ideal Gas Constant)

Calculate the number of moles (n) using the Ideal Gas Law

Use the rearranged Ideal Gas Law formula to solve for n: \(n = \frac{PV}{RT}\) Plug in the given values and calculate n: \(n = \frac{(1.97 \mathrm{atm})(6.0 \mathrm{L})}{(0.0821 \frac{\mathrm{Latm}}{\mathrm{Kmol}}) (298 \mathrm{K})} = 0.483 \mathrm{mol}\) So, 10 meters below water, there are 0.483 moles of air in the lungs. #c. At the top of Mount Everest (T=200 K, P=0.296 atm)#

Write down the given variables

For this situation, we have: - P = 0.296 atm - V = 6.0 L - T = 200 K - R = 0.0821 Latm / Kmol (Ideal Gas Constant)

Calculate the number of moles (n) using the Ideal Gas Law

Use the rearranged Ideal Gas Law formula to solve for n: \(n = \frac{PV}{RT}\) Plug in the given values and calculate n: \(n = \frac{(0.296 \mathrm{atm})(6.0 \mathrm{L})}{(0.0821 \frac{\mathrm{Latm}}{\mathrm{Kmol}}) (200 \mathrm{K})} = 0.108 \mathrm{mol}\) So, at the top of Mount Everest, there are 0.108 moles of air in the lungs.

Key concepts

Moles Calculation

When trying to find how many moles of gas are in your lungs in varying situations, you use the Ideal Gas Law. This law is represented by the equation \[ PV = nRT \]where

• \( P \) is the pressure in atmospheres (atm),
• \( V \) is the volume in liters (L),
• \( n \) is the number of moles,
• \( R \) is the Ideal Gas Constant, and
• \( T \) is the temperature in Kelvin (K).

To find the number of moles \( n \), you rearrange the formula to\[ n = \frac{PV}{RT} \]Plugging in the given values allows calculation of \( n \) for different conditions. The process involves substituting the pressure, volume, temperature, and using the constant value to solve for the number of moles. Each scenario simply requires plugging in the distinct conditions of pressure and temperature.

Gas Constants

Understanding gas constants is crucial for applying the Ideal Gas Law effectively. The gas constant \( R \) is a fixed value used to correlate the physical properties of gases. For ideal gas calculations in standard conditions, the value of \( R \) is usually \[ R = 0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{K} \cdot \mathrm{mol}} \]This constant allows you to bridge the units of pressure, volume, and temperature to find the number of moles.Remember, the Ideal Gas Law assumes perfect gas behavior which is mostly accurate under standard conditions of temperature and pressure, unless the gas undergoes extreme temperatures or pressures where it might deviate from ideal behavior.

Pressure and Temperature Change

Gas behavior significantly depends on changes in pressure and temperature. When the pressure increases, as it would underwater, the number of moles of gas in a confined space increases if volume and temperature are consistent. For instance, in the underwater scenario at 1.97 atm versus the sea level situation at 1.00 atm, while the volume and temperature remain the same, more moles of air are present in the lungs due to higher pressure. Similarly, temperature variations also affect physical properties. Higher temperatures tend to increase the kinetic energy of the gas molecules, stimulating expansion and potentially increasing volume if pressure is constant. At the top of Mount Everest with a lower temperature of 200 K and pressure of 0.296 atm, fewer moles of air exist in the lungs due to the reduced kinetic energy and expansion capacity of the gas. This highlights how pressure and temperature interplay affects the number of moles calculated for different environments.