Problem 59 - Chemistry The Central Science In Si Units

Question

During a person's typical breathing cycle, the \(\mathrm{CO}_{2}\) concentration in the expired air rises to a peak of \(4.6 \%\) by volume. (a) Calculate the partial pressure of the \(\mathrm{CO}_{2}\) in the expired air at its peak, assuming \(101.3 \mathrm{kPa}\) pressure and a body temperature of \(37^{\circ} \mathrm{C}\). (b) What is the molarity of the \(\mathrm{CO}_{2}\) in the expired air at its peak, assuming a body temperature of \(37^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

CO2 partial pressure: 4.66 kPa; Molarity: 0.181 mol/L.

Step 1: Calculate the Partial Pressure of CO2

To find the partial pressure of CO2 in expired air, use the formula for partial pressure, which is given by \( P_{\text{partial}} = \frac{x}{100} \times P_{\text{total}} \). Given that the concentration of CO2 is 4.6% by volume and the total pressure \( P_{\text{total}} \) is 101.3 kPa, we substitute these values into the equation: \[ P_{\text{CO}_2} = \frac{4.6}{100} \times 101.3 \text{ kPa} = 4.6598 \text{ kPa} \] The partial pressure of CO2 is approximately 4.66 kPa.

Step 2: Convert Temperature to Kelvin

Before calculating molarity, convert the given body temperature from Celsius to Kelvin. The conversion formula is \( T_{\text{K}} = T_{\text{C}} + 273.15 \). For a body temperature of 37°C, the conversion is: \[ T_{\text{K}} = 37 + 273.15 = 310.15 \text{ K} \]

Step 3: Calculate Molar Volume

Determine the molar volume of a gas at body temperature using the ideal gas law \( PV = nRT \). Here, \( R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \), and \( P \) is 101.3 kPa or 101,300 Pa. The molar volume \( V_m \) is calculated by rearranging the equation to \( V_m = \frac{RT}{P} \). Substitute \( R = 8.314 \), \( T = 310.15 \), and \( P = 101300 \): \[ V_m = \frac{8.314 \times 310.15}{101300} = 0.0254 \text{ m}^3 \text{ per mol} \] which converts to 25.4 liters per mole.

Step 4: Calculate Molarity of CO2

Molarity \( C \) is defined as moles of solute per liter of solution. Since the molar volume has been calculated, we can find the molarity using the partial pressure from Step 1. The formula is \( C = \frac{P_{\text{CO}_2}}{RT} \):\[ C = \frac{4.6598 \times 10^3}{8.314 \times 310.15} \approx 0.181 \text{ mol/L} \] Therefore, the molarity of CO2 in the expired air at its peak is approximately 0.181 mol/L.

Key Concepts

Partial Pressure

Partial pressure is a way to express the pressure exerted by a single gas in a mixture of gases. This is especially useful when dealing with gases like carbon dioxide (0CO\textsubscript{2}) in the complex mixture that is air. In a given container, different gases will exert their individual pressures, known as partial pressures. The sum of these partial pressures equals the total pressure in the system.
When calculating the partial pressure of a gas, the equation used is:
\[P_{\text{partial}} = \frac{x}{100} \times P_{\text{total}} \] where \( x \) is the percentage concentration of the gas and \( P_{\text{total}} \) is the total pressure of the gas mixture.
For instance, if we have a mixture of gases under a pressure of 101.3 kPa, and \( 4.6\% \) of this mixture is \( CO\textsubscript{2} \), then the calculation would be:
\[P_{CO_2} = \frac{4.6}{100} \times 101.3 \text{ kPa} = 4.6598 \text{ kPa} \] This calculation shows that the \( CO\textsubscript{2} \) exerts a partial pressure of approximately 4.66 kPa. Partial pressure is a fundamental concept of gas laws, critical for fields ranging from chemistry to medicine where gases are in mixtures.

Ideal Gas Law

The ideal gas law is an essential equation in chemistry and physics that describes the relationship between the pressure, volume, temperature, and amount of an ideal gas. It is represented as:
\[PV = nRT\]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (8.314 J mol\(^{-1}\) K\(^{-1}\)), and \( T \) is the temperature in Kelvin.
This equation assumes a perfect scenario where the gas particles do not interact and occupy no volume. While no real gases behave perfectly according to the ideal gas law, many gases at low pressure and high temperature approximate it closely enough for practical purposes. It is a useful tool for calculating the behavior of gases under varying conditions.

Molar Volume

Molar volume is the volume occupied by one mole of a substance, typically a gas under specified conditions of temperature and pressure. At standard temperature and pressure (STP), the molar volume of an ideal gas is about 22.4 liters per mole.
However, conditions can vary, such as at body temperature and pressure (like in the human lungs), requiring the ideal gas law to adjust the calculation. To find the molar volume of a gas at any given temperature and pressure, the formula \( V_m = \frac{RT}{P} \) is used, where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( P \) is the pressure in Pascal.
In real-world applications such as respiratory physiology, understanding molar volume helps determine how gases behave in biological systems. For example, the molar volume at body temperature and standard pressure was found to be 25.4 liters per mole, reflecting a more specific environment than the generic STP.

Molarity

Molarity is a measure of the concentration of a solute in a solution. Specifically, it is defined as the number of moles of solute per liter of solution. It is a common and practical unit of concentration used in chemistry to express how much of a given substance is present in a mixture.

• Molarity is expressed in units of moles per liter (mol/L).
• To calculate molarity, divide the amount of solute by the volume of the solution in liters.

In the context of the given exercise, the molarity of
\( CO\textsubscript{2} \) in expired air can be calculated using the formula:
\[ C = \frac{P_{\text{CO}_2}}{RT} \]
Where \( P_{\text{CO}_2} \) is the partial pressure, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This way, we determined that the molarity of \( CO\textsubscript{2} \) in the expired air at peak concentration is approximately 0.181 mol/L. Molarity is crucial for understanding how gases like carbon dioxide dissolve and react in biological systems.