Question

A mixture of cyclopropane and oxygen is sometimes used as a general anesthetic. Consider a balloon filled with an anesthetic mixture of cyclopropane and oxygen at \(170 .\) torr and 570 . torr. respectively. Calculate the ratio of the moles \(\mathrm{O}_{2}\) to moles cyclopropane in this mixture.

Short answer

The ratio of moles of oxygen to cyclopropane in the mixture is 3.35:1.

Step-by-step solution

Understand Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. Mathematically, this can be represented as: \(P_{total} = P_{1} + P_{2} + P_{3} + ...\), where P is the pressure and subscript numbers represent different gases.

Use given partial pressures to find total pressure

In our case, we are given the partial pressures of cyclopropane (\(P_{Cyclo}\)) and oxygen (\(P_{O_2}\)). We can find the total pressure using Dalton's law: \(P_{total} = P_{Cyclo} + P_{O_2}\) \(P_{total} = 170 \, \mathrm{torr} + 570 \, \mathrm{torr}\) \(P_{total} = 740 \, \mathrm{torr}\)

Find the mole fraction of each gas in the mixture

The mole fraction is the ratio of the partial pressure of a gas to the total pressure of the mixture. We will find the mole fractions of oxygen (\(X_{O_2}\)) and cyclopropane (\(X_{Cyclo}\)): \(X_{O_2} = \frac{P_{O_2}}{P_{total}}\) \(X_{Cyclo} = \frac{P_{Cyclo}}{P_{total}}\) Plug in the known values to calculate the mole fractions: \(X_{O_2} = \frac{570 \, \mathrm{torr}}{740 \, \mathrm{torr}}\) \(X_{Cyclo} = \frac{170 \, \mathrm{torr}}{740 \, \mathrm{torr}}\)

Calculate the ratio of moles for oxygen to cyclopropane

To calculate the molar ratio of oxygen to cyclopropane in the mixture, we will simply divide the mole fraction of oxygen by the mole fraction of cyclopropane: Molar Ratio (\(O_2:Cyclo\)) = \(\frac{X_{O_2}}{X_{Cyclo}}\) Substitute the values of mole fractions: Molar Ratio (\(O_2:Cyclo\)) = \(\frac{\frac{570}{740}}{\frac{170}{740}}\) Simplify: Molar Ratio (\(O_2:Cyclo\)) = \(\frac{570}{170}\) Molar Ratio (\(O_2:Cyclo\)) = 3.35 The ratio of moles of oxygen to cyclopropane in the mixture is 3.35:1.

Key concepts

Gas Mixtures

In many scientific applications, such as the example of anesthetic mixtures, understanding gas mixtures is essential. A gas mixture consists of different types of gases combined together. Despite being a blend of different gases, each gas within the mixture behaves as though it occupies the entire volume on its own. This behavior leads to the need to evaluate each gas's contribution to the total properties of the mixture, such as pressure. Daltons’s Law of Partial Pressures is particularly useful in comprehending how these mixtures behave, as it allows us to calculate the total pressure of the mixture from the individual pressures of each gas. This foundational understanding is crucial when working with gas mixtures in fields like medicine and engineering.

Mole Fraction

The mole fraction is a key concept when dealing with gas mixtures. It represents the ratio or fraction of the total moles represented by a particular component of the mixture. This concept is important because it provides insight into how much of the mixture is made up of each gas.
\(X\), the symbol for mole fraction, is calculated by dividing the partial pressure of the gas by the total pressure of the mixture.
In this case, for oxygen, the mole fraction can be calculated by using the formula:

• \(X_{O_2} = \frac{P_{O_2}}{P_{total}}\)

This gives us a clear perspective on how much each gas contributes to the overall mixture. It can be particularly useful in many calculations involving gases and their interactions.

Partial Pressure

Partial pressure is quite an intriguing aspect of gas mixtures. It refers to the pressure one component of a gas mixture would exert if it occupied the whole volume by itself. Each gas in a mixture exerts pressure independently, and Dalton’s Law helps quantify how each contributes to the total pressure. Mathematically, this is expressed for each component as:

• \(P_i = X_i \times P_{total}\)

where \(P_i\) is the partial pressure of the component, \(X_i\) its mole fraction, and \(P_{total}\) the total pressure.
For example, oxygen in the mixture has a partial pressure of 570 torr. This value contributes to determining the mole fraction and the component's significance within the mixture. Understanding partial pressures can help predict how gases behave in different environments, and this principle is crucial in various fields, including chemistry and physics.

Molar Ratio

The molar ratio in a gas mixture identifies the relationship between the amounts of different gases present. It's calculated by taking the mole fractions of the gases and comparing them.
The importance of calculating the molar ratio lies in its ability to show how quantities are distributed among different gases. For instance, in our exercise, the molar ratio of oxygen to cyclopropane was determined using the formula:

• Molar Ratio \((O_2: Cyclopropane) = \frac{X_{O_2}}{X_{Cyclo}}\)

This ratio was calculated as 3.35:1, indicating that for every mole of cyclopropane, there are 3.35 moles of oxygen. Such insights are particularly valuable in chemical reactions and industrial processes, where knowing the proportions of reactants can significantly influence the outcome.