Problem 86 - Chemistry The Molecular Nature of Matter

Question

A mixture of 26,000 torr \(\mathrm{CO}_{2}\) and 104,000 torr \(\mathrm{N}_{2}\) is sold for packaging food. What are the mole fraction and mole percent of \(\mathrm{CO}_{2}\) in this mixture?

Step-by-Step Solution

Answer

The mole fraction of \(\mathrm{CO}_2\) is 0.2, and the mole percent is 20%.

Step 1: Calculate Total Pressure

Sum the partial pressures of \(\mathrm{CO}_2\) and \(\mathrm{N}_2\) to find the total pressure of the mixture. Total pressure = pressure of \(\mathrm{CO}_2\) + pressure of \(\mathrm{N}_2\).

Step 2: Calculate Mole Fraction of \(\mathrm{CO}_2\)

Use the partial pressure of \(\mathrm{CO}_2\) and the total pressure to find the mole fraction of \(\mathrm{CO}_2\). Mole fraction of \(\mathrm{CO}_2\) = partial pressure of \(\mathrm{CO}_2\) / total pressure.

Step 3: Calculate Mole Percent of \(\mathrm{CO}_2\)

Multiply the mole fraction of \(\mathrm{CO}_2\) by 100 to convert it to a percentage. Mole percent = mole fraction of \(\mathrm{CO}_2\) \(\times\) 100.

Key Concepts

Partial Pressures in Mixtures

In chemistry, when dealing with gas mixtures, it's important to understand the behavior of individual gases as part of a whole. Each gas in a mixture exerts a pressure as if it were alone in the container. This is known as the gas's partial pressure. The idea is quite similar to how each person in a group project contributes a part of the work.

For example, suppose you have a container with both oxygen and nitrogen gases. The overall pressure inside the container is the sum of the pressure from oxygen and the pressure from nitrogen. Each of these pressures is considered 'partial' because it represents only a part of the total pressure. This becomes particularly useful when gases do not react with each other and retain their individual properties despite being in a mix.

Mole Fraction Calculation

The mole fraction is a way of expressing the concentration of a component in a mixture. Think of it like figuring out what slice of the pie belongs to a particular flavor in a mixed fruit pie. To calculate mole fraction, you don't need actual amounts, just the ratio.

Here's how to work it out: You divide the number of moles (or in the case of gases under the same conditions, partial pressures can be used as a proxy for moles) of one component by the total number of moles (or total partial pressure) in the mixture. It's a unitless number that explains the proportion of a component relative to the entire mixture.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is named after the English scientist John Dalton, who formulated it. This principle states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

Imagine you have a party balloon filled with both helium and oxygen. Dalton's Law tells us that the pressure inside the balloon is simply the addition of the pressure from helium plus the pressure from oxygen. It's as if each gas is acting alone and doesn't 'know' the others are there. This law is particularly handy when studying gas mixtures because it lets us predict the total pressure from the pressures of its parts.