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Chapter 13: Problem 56

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

Expert verified
The concentration of CO鈧� in air in terms of (a) mol percentage is 4%, and (b) molarity is 0.0016 M at 1 atm pressure and 37掳C.

Step by step solution

01

(a) Find the mol percentage of CO鈧� in air

1. We are given that the \% by volume of CO鈧� 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 CO鈧� in the air is 4%.
02

(b) Find the molarity of CO鈧� in air

To calculate the molarity of CO鈧� 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 CO鈧� in 1 L of air. We know that 4% of the air by volume is occupied by CO鈧�. So, in 1 L of air, CO鈧� 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 CO鈧�) Temperature (T) = 37掳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 CO鈧� (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 CO鈧� in air. Since we assumed 1 L of air, the molarity of CO鈧� (moles per liter) can be calculated directly from the moles of CO鈧�. Molarity = \( \frac{0.0016 \, \mathrm{mol}}{1 \, \mathrm{L}}\) = 0.0016 M So, the concentration of CO鈧� in air in terms of (a) mol percentage is 4%, and (b) molarity is 0.0016 M at 1 atm pressure and 37掳C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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% CO鈧�, 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 CO鈧� 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 CO鈧� can affect weather patterns and have health implications for humans and animals.
  • Normal levels of CO鈧� 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.

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Most popular questions from this chapter

The presence of the radioactive gas radon \((\mathrm{Rn})\) in well water obtained from aquifers that lie in rock deposits presents a possible health hazard in parts of the United States. (a) Assuming that the solubility of radon in water with 1 atm pressure of the gas over the water at \(30^{\circ} \mathrm{C}\) is \(7.27 \times 10^{-3} \mathrm{M}\), what is the Henry's law constant for radon in water at this temperature? (b) A sample consisting of various gases contains \(3.5 \times 10^{-6}\) mole fraction of radon. This gas at a total pressure of \(32 \mathrm{~atm}\) is shaken with water at \(30^{\circ} \mathrm{C}\). Calculate the molar concentration of radon in the water.

A textbook on chemical thermodynamics states, "The heat of solution represents the difference between the lattice energy of the crystalline solid and the solvation energy of the gaseous ions." (a) Draw a simple energy diagram to illustrate this statement. (b) A salt such as NaBr is insoluble in most polar nonaqueous solvents such as acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) or nitromethane \(\left(\mathrm{CH}_{3} \mathrm{NO}_{2}\right)\), but salts of large cations, such as tetramethylammonium bromide \(\left[\left(\mathrm{CH}_{3}\right)_{4} \mathrm{NBr}\right]\), are generally more soluble. Use the thermochemical cycle you drew in part (a) and the factors that determine the lattice energy (Section 8.2) to explain this fact.

Acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) is a polar organic solvent that dissolves a wide range of solutes, including many salts. The density of a \(1.80 \mathrm{M}\) LiBr solution in acetonitrile is \(0.826 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the concentration of the solution in (a) molality, (b) mole fraction of LiBr, (c) mass percentage of \(\mathrm{CH}_{3} \mathrm{CN}\).

(a) What is an ideal solution? (b) The vapor pressure of pure water at \(60^{\circ} \mathrm{C}\) is 149 torr. The vapor pressure of water over a solution at \(60^{\circ} \mathrm{C}\) containing equal numbers of moles of water and ethylene glycol (a nonvolatile solute) is 67 torr. Is the solution ideal according to Raoult's law? Explain.

(a) Would you expect stearic acid, \(\mathrm{CH}_{3}\left(\mathrm{CH}_{2}\right)_{16} \mathrm{COOH}\), to be more soluble in water or in carbon tetrachloride? Explain. (b) Which would you expect to be more soluble in water, cyclohexane or dioxane? Explain.
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