Question

Breathing air that contains \(4.0 \%\) by volume \(\mathrm{CO}_{2}\) over time causes rapid breathing, throbbing headache, and nausea, among other symptoms. What is the concentration of \(\mathrm{CO}_{2}\) in such air in terms of (a) mol percentage, (b) molarity, assuming 1 atm pressure and a body temperature of \(37{ }^{\circ} \mathrm{C} ?\)

Short answer

The concentration of CO₂ in air with 4% by volume CO₂ is (a) 4% mol percentage, and (b) approximately 0.00156 mol/L molarity, assuming 1 atm pressure and a body temperature of 37°C.

Step-by-step solution

Convert the percentage by volume to mol percentage

Given that the percentage by volume of CO₂ is 4 %, we can directly say that the mol percentage of CO₂ is also 4 %, because the volume ratios of gases will remain the same as the mole ratios at the given temperature and pressure (according to Avogadro's law).

Convert the temperature from Celsius to Kelvin

The given body temperature is 37 °C. In order to use the ideal gas law, we need to convert the temperature to Kelvin. We can do this using the following formula: \(T(K) = T(°C) + 273.15\) Plugging in the given temperature, \(T(K) = 37 °C + 273.15 = 310.15 K \)

Calculate the partial pressure of CO₂

Knowing the mol percentage of CO₂ (4 %) and the total pressure (1 atm), we can calculate the partial pressure of CO₂ by multiplying the mol percentage by the total pressure: \(P_{CO_2} = \text{mol percentage of } CO_2 \times \text{total pressure}\) \(P_{CO_2} = 0.04 \times 1 \,atm = 0.04 \,atm\)

Apply the Ideal Gas Law to find the molarity of CO₂

The ideal gas law equation is: \[ PV = nRT\] where P is the pressure (in atm), V is the volume (in L), n is the number of moles, R is the ideal gas constant (0.0821 L×atm/mol×K), and T is the temperature (in K). We want to find the molarity, which is defined as the number of moles (n) per volume (V) of the solution (in mol/L). We can rearrange the ideal gas law equation: \[\frac{n}{V} = \frac{P}{RT}\] Now, plug in the pressure of CO₂ (0.04 atm), the temperature in Kelvin (310.15 K), and the gas constant R (0.0821 L×atm/mol×K) into the equation: \[\frac{n}{V} = \frac{0.04 \,atm}{(0.0821 \,L×atm/mol×K) \times (310.15 \,K)} \] Solve the equation: \[\frac{n}{V} \approx 0.00156\,\frac{mol}{L}\] So, the molarity of CO₂ is approximately 0.00156 mol/L. To summarize, the concentration of CO₂ in such air is (a) 4 % mol percentage (b) 0.00156 mol/L molarity

Key concepts

Mole Percentage

Mole percentage is an expression of the concentration of a component in a mixture. It is defined as the amount of moles of a substance compared to the total moles of all substances present, multiplied by 100 to get a percentage. In the context of gases, if a gas mixture contains 4.0% CO2 by volume at constant temperature and pressure, it means that 4.0% of the total moles of gas present are CO2. This is based on Avogadro's law, which states that equal volumes of gases at the same temperature and pressure contain an equal number of moles.

In calculations, mole percentage is straightforward to use, especially when dealing with gas mixtures, since one can directly equate volume percentage to mole percentage. Therefore, in the given exercise, 4.0% by volume translates directly to a mole percentage of 4.0%.

Molarity

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters (\frac{moles}{liters}). This concept is extremely useful in chemistry as it allows for the easy calculation of reactants and products in chemical reactions.

Calculating Molarity from the Ideal Gas Law

When applied to gases, if we know the partial pressure of a gas, its molarity can be calculated using the ideal gas law with the formula:
\[ \frac{n}{V} = \frac{P}{RT} \]
where P is the pressure of the gas, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. By rearranging the ideal gas law, one can solve for molarity, making it a central concept in gas concentration calculations as demonstrated in the CO2 concentration problem.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas. The equation is:
\[ PV = nRT \]
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

This law is a combination of Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac's law and is particularly useful for predicting the behavior of gases under different conditions. However, it assumes that gases are ideal, which means that it neglects interactions between gas molecules and assumes that the gas molecules themselves take up no space. Though real gases do not always follow this law exactly, it is a good approximation in many conditions, such as at high temperature and low pressure.

Avogadro's Law

Avogadro's law states that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. It can be simply expressed as:
\[ \frac{V}{n} = k \]
where V is the volume of the gas, n is the number of moles, and k is a constant for a given temperature and pressure.

Avogadro's law implies that the volume of gas is directly proportional to the number of moles of gas when temperature and pressure are held constant. This law is critical in understanding the relationship between moles and volume in gases and is the basis for determining the molar volume of a gas. It is also instrumental when relating volume percentages to mole percentages in gas mixtures, as was crucial in the CO2 example provided in the exercise.

Partial Pressure

Partial pressure is the pressure that a gas would exert if it alone occupied the entire volume of a mixture at the same temperature. Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

Importance in Gas Concentration Calculations

The partial pressure is important in calculations involving gas mixtures because it allows for the determination of the concentration of individual gases within the mix. When combined with the ideal gas law, it enables the calculation of molarity, as shown in the problem where knowing the partial pressure of CO2 was essential to find its molarity in the mixture at body temperature. This concept is regularly used in various scientific fields, including atmospheric science, respiratory physiology, and anesthesiology, to understand gas compositions and effects.