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 in terms of (a) mol percentage is 4%, and (b) molarity is 0.0016 M at 1 atm pressure and 37°C.

Step-by-step solution

(a) Find the mol percentage of CO₂ in air

1. We are given that the \% by volume of CO₂ in air is 4%. Since air is a mixture of gases, we assume it behaves like an ideal gas. So, the `%` by volume is equal to the `%` by moles. 2. Hence, mol percentage of CO₂ in the air is 4%.

(b) Find the molarity of CO₂ in air

To calculate the molarity of CO₂ in the air, we will use the Ideal Gas Law, given by: \(PV = nRT\) First, let's assume that we are working with 1 liter (L) of air. 1. Calculate the moles of CO₂ in 1 L of air. We know that 4% of the air by volume is occupied by CO₂. So, in 1 L of air, CO₂ occupies 0.04 L (4% of 1 L). 2. Now, write down the Ideal Gas Law variables at the given conditions. Pressure (P) = 1 atm Volume (V) = 0.04 L (volume occupied by CO₂) Temperature (T) = 37°C = 310 K (convert to Kelvin) Using the universal gas constant (R) = 0.0821 L atm / (mol K) 3. With the given variables, we can now solve the Ideal Gas Law for the moles of CO₂ (n). \(n = \frac{PV}{RT}\) \(n = \frac{(1 \, \mathrm{atm})(0.04 \, \mathrm{L})}{(0.0821 \, \mathrm{L\,atm\,K^{-1}\,mol^{-1}})(310 \, \mathrm{K})}\) \(n \approx 0.0016 \, \mathrm{mol}\) 4. Calculate the molarity of CO₂ in air. Since we assumed 1 L of air, the molarity of CO₂ (moles per liter) can be calculated directly from the moles of CO₂. Molarity = \( \frac{0.0016 \, \mathrm{mol}}{1 \, \mathrm{L}}\) = 0.0016 M So, the concentration of CO₂ in air in terms of (a) mol percentage is 4%, and (b) molarity is 0.0016 M at 1 atm pressure and 37°C.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of gases under set conditions of pressure, volume, and temperature. This law states that for a given quantity of gas, the equation \(PV = nRT\) holds true. Here, \(P\) represents the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant (0.0821 L atm / (mol K)), and \(T\) is the temperature in Kelvin.

Understanding the Ideal Gas Law is crucial because it allows us to predict how gases will react to changes in their environment. For instance, if the temperature or pressure of the gas changes, we can calculate a new volume or number of moles.

This law assumes that gas particles are very small compared to the space they occupy and that they are in constant, random motion. Although real gases differ slightly, the Ideal Gas Law gives a close approximation for many situations, especially at high temperatures and low pressures.

Molarity calculation

Molarity is a measure of the concentration of a solute within a solution and is expressed in moles per liter (M). To calculate molarity, you need to know the amount of solute in moles and the volume of the solution in liters.

In exercises involving gases, we often use the Ideal Gas Law to find the number of moles from given conditions. For example, if we're considering 1 liter of air with 4% CO₂, we calculate the moles of gas using volume data and the Ideal Gas Law to derive a molarity.

• Convert any given temperatures to Kelvin by adding 273 to the Celsius reading.
• Use pressure in atmospheres (if needed, convert from other units).
• Once the moles are determined with \(n = \frac{PV}{RT}\), divide by the volume of the solution, usually assumed as 1 L, to find molarity.

Understanding molarity helps us measure and understand how much of a substance is present in a particular space, which is important in scenarios like determining the concentrations that affect bodily functions.

Chemical concentrations

Chemical concentration refers to the amount of a substance contained in a certain volume. Knowing concentrations is vital in chemistry and medicine because it can affect reactions and biological processes.

Concentration can be expressed in various ways, such as:

• Volume percentage: Like in our exercise, where CO₂ comprises 4% by volume of air.
• Mol percentage: Which will be the same as the volume percentage for gases behaving ideally.
• Molarity: Which combines volume and molar data to give concentration in terms of moles per liter.

Each method has its applications depending on the context, such as industrial chemical processes or biological systems. In our exercise, transforming between these forms helps us understand how exposure levels correlate with physiological effects like headaches.

Air composition

Air is a mixture of different gases, primarily nitrogen (about 78%), oxygen (about 21%), and small amounts of other gases including carbon dioxide, argon, and water vapor.

Understanding air composition is important as even small changes can have significant effects. For instance, increases in CO₂ can affect weather patterns and have health implications for humans and animals.

• Normal levels of CO₂ in air are around 0.04%, but our exercise checks effects at a much higher concentration (4%).
• These increased levels can cause symptoms due to the body's sensitivity to changes in gas concentrations, affecting respiratory and cognitive function.
• Assessing air composition helps in environmental monitoring and making decisions to safeguard health and ecosystems.

By analyzing such compositions, scientists can foretell potential hazards and recommend safety measures or limits for different environments.