Problem 69 - Chemistry

Question

Dry air near sea level has the following composition by volume: \(\mathrm{N}_{2}, 78.08\) percent; \(\mathrm{O}_{2}, 20.94\) percent; Ar, 0.93 percent; \(\mathrm{CO}_{2}, 0.05\) percent. The atmospheric pressure is 1.00 atm. Calculate (a) the partial pressure of each gas in atm and (b) the concentration of each gas in moles per liter at \(0^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The partial pressures (atm) of \(\mathrm{N}_{2}\), \(\mathrm{O}_{2}\), Ar, and \(\mathrm{CO}_{2}\) are approximately 0.7808, 0.2094, 0.0093, and 0.0005 respectively. The molar concentrations (mol/L) at \(0^{\circ} \mathrm{C}\) of \(\mathrm{N}_{2}\), \(\mathrm{O}_{2}\), Ar, and \(\mathrm{CO}_{2}\) are correspondingly approximately 0.031, 0.008, 0.0004, and 0.00002. Note: results depend on the given values in the problem

Step 1: Compute the Partial Pressures

The partial pressure of each gas can be calculated using the percentage of the volume occupied by each individual gas. The volume percentage is equal to the pressure percentage. Thus, multiply each percentage by the total pressure of 1.00 atm. Example: To find the partial pressure of \(\mathrm{N}_{2}\), multiply its percentage, 78.08% by 1.00 atm:

Step 2: Compute Molar Concentration

To calculate the molar concentration, use the Ideal Gas Law, \(PV=nRT\), but rearrange it to solve for \(n/V\) which is the definition of molarity (M). Thus, the equation becomes \(M = P/RT\). Since R is the ideal gas constant with a value of 0.08206 L*atm/(mol*K), T is the temperature in kelvin (which is 273.15 for \(0^{\circ} \mathrm{C}\)), you can compute for molarity M for each gas by substituting the respective partial pressures computed from Step 1.

Steps 3-6 Repeat for Other Gases

Repeat steps 1 and 2 for the other gases: \(\mathrm{O}_{2}\), Ar, and \(\mathrm{CO}_{2}\). Use their respective percentages, 20.94%, 0.93% and 0.05% to find their partial pressures and molar concentrations respectively.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates four key properties of gases: pressure (P), volume (V), amount (n), and temperature (T). This law is expressed as \(PV = nRT\), where \(R\) is the ideal gas constant. This relationship helps predict the behavior of gases under different conditions.

For our problem involving the calculation of molar concentration, we need to rearrange the Ideal Gas Law. We focus on the volume and amount of gas which gives us molarity \(M = \frac{n}{V}\), thus leading to \(M = \frac{P}{RT}\).

• P is the partial pressure of the gas in atm.
• R is typically 0.08206 L\(\cdot\)atm/(mol\(\cdot\)K).
• T is the temperature in Kelvin, and for 0°C, T becomes 273.15 K.

This relationship becomes crucial when calculating the molar concentration of gases, especially when starting with their partial pressures.

Molar Concentration

Molar concentration, or molarity, measures how many moles of a substance are present in a liter of solution. When considering gases, molar concentration is calculated using the derived formula \(M = \frac{P}{RT}\). This formula originates from rearranging the Ideal Gas Law.

To compute the molar concentration of each gas mentioned, take the individual partial pressures and apply these values. This measure is significant in science as it helps to quantify reactions involving gases. Understanding molar concentration:

• Allows for the comparison of different gas ratios in chemical reactions.
• Facilitates predictions about the behavior and changes of gases in a mixture.

By calculating the molar concentration, we gain insight into the potential reactivity of the gases present in the air.

Atmospheric Pressure

Atmospheric pressure is the weight of the air above a given point. At sea level, it is commonly accepted as 1 atm (atmosphere). This standard pressure serves as a reference point in many scientific calculations and experiments. When dealing with problems involving gases, the pressure of the atmosphere is key to calculating partial pressures.

In this problem, the total atmospheric pressure is 1.00 atm, and understanding this enables us to calculate the partial pressures of the individual gases. Some points about atmospheric pressure:

• It can affect the boiling and melting points of substances.
• It varies with altitude, influencing how we perceive air pressure at different heights.

For gases, the atmospheric pressure demonstrates the overall contribution of all gas molecules in the air, which is vital when deciphering the behavior of each gas present.

Volume Composition of Gases

The volume composition of a gas mixture gives information about the proportions of each gas present in the mixture. In dry air at sea level, the primary constituents are nitrogen (78.08%), oxygen (20.94%), argon (0.93%), and carbon dioxide (0.05%). These percentages indicate the volume occupied by each gas in a mix and directly correspond to their respective partial pressures.

The volume percentage of a gas is essential in determining its behavior within a mixture. In calculations, each percentage translates to the same percentage of total pressure. For example, if you take nitrogen's 78.08%, its partial pressure will be 78.08% of the total atmospheric pressure of 1 atm. Important aspects of volume composition include:

• It helps in calculating gas contributions to phenomena like air quality.
• Volume compositions reflect directly in techniques like gas chromatography.

Understanding the volume composition is key to insights into the role each gas plays in the atmosphere.