Problem 6 - Chemistry The Molecular Science

Question

State Dalton's law of partial pressures. If the air we breathe is \(78 \% \mathrm{~N}_{2}\) and \(21 \% \mathrm{O}_{2}\) on a mole basis, calculate the mole fraction of \(\mathrm{O}_{2}\). Calculate the partial pressure of \(\mathrm{O}_{2}\) if the total pressure is \(720 \mathrm{mmHg}\).

Step-by-Step Solution

Answer

The partial pressure of \( \text{O}_2 \) is 151.2 mmHg.

Step 1: Understand Dalton's Law of Partial Pressures

Dalton's law of partial pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each constituent gas. Mathematically, it can be expressed as \( P_{total} = P_{1} + P_{2} + ext{...} + P_{n} \), where each \( P_{i} \) represents the partial pressure of a gas in the mixture.

Step 2: Determine the Mole Fraction of Oxygen

The mole fraction of a gas in a mixture is the ratio of the number of moles of that gas to the total number of moles of all gases present. Given that the air is composed of \( 78\% \) \( ext{N}_2 \) and \( 21\% \) \( ext{O}_2 \), the mole fraction of \( ext{O}_2 \) is \( \frac{21}{100} = 0.21 \).

Step 3: Calculate the Partial Pressure of Oxygen

Using Dalton's law, the partial pressure of \( ext{O}_2 \) can be found by multiplying its mole fraction by the total pressure. Therefore, the partial pressure of \( ext{O}_2 \) is \( P_{ ext{O}_2} = 0.21 \times 720 \text{ mmHg} = 151.2 \text{ mmHg} \).

Key Concepts

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture and is particularly useful in gas mixtures. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components present in the mixture.
This simple but important concept helps us understand the proportion of different gases in a mixture without needing to know their actual mass or volume.Calculating the mole fraction involves the following steps:

• Identify the number of moles of each gas in the mixture.
• Add them up to find the total number of moles.
• Take the ratio of the number of moles of the gas in question to the total moles.

For example, if air is composed of 21% oxygen on a mole basis, then the mole fraction of oxygen in the air would be \( \frac{21}{100} = 0.21 \). This calculation helps us quantify the presence of oxygen among other gases.

Partial Pressure Calculation

Partial pressure is the pressure exerted by a single component of a gas mixture if it occupied the entire volume on its own, at the same temperature. Dalton's Law of Partial Pressures helps us find this by using the mole fraction and total pressure.To calculate partial pressure:

• First, calculate the mole fraction of the gas you're interested in. This was found to be 0.21 in the previous section for oxygen.
• Next, multiply the mole fraction by the total pressure of the gas mixture.

This translates mathematically to \( P_{gas} = \text{mole fraction} \times P_{total} \).
For example, if the total pressure of the air is 720 mmHg and the mole fraction of oxygen is 0.21, the partial pressure of oxygen would be \( 0.21 \times 720 \text{ mmHg} = 151.2 \text{ mmHg} \). This process allows us to determine how each gas contributes to the overall pressure of the mixture.

Gas Mixtures Pressure

Gas mixtures pressure is the combined pressure of all gases present within a single container. It's not simply the pressure of the most abundant gas, but rather, each type of gas makes its individual contribution to the total pressure.Dalton's Law of Partial Pressures illustrates this by stating that the total pressure of a mixture is the sum of all individual partial pressures of each gas component:\[ P_{total} = P_1 + P_2 + \, \ldots \, + P_n \]This reveals the collaborative nature of gases, where each contributes based on its quantity and identity, irrespective of their chemical properties.
For instance, even if nitrogen (N₂) makes up 78% of air, the pressure contribution from oxygen (O₂) at 21% is calculated separately and then added to nitrogen's part.
This cumulative approach allows chemists and scientists to predict how gas mixtures will behave under different conditions, crucial for numerous applications in science and engineering.