Question

A 1;1 mixture by moles of nitrous oxide and oxygen is often used as a sedative in dentistry. If the total pressure of this mixture in a cylinder is 2.50 atm, what is the partial pressure of each gas?

Short answer

The partial pressure of nitrous oxide (N2O) and oxygen (O2) in the cylinder is 1.25 atm for each gas.

Step-by-step solution

Understand Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases. Mathematically, this is represented as: \[P_{total} = P_{1} + P_{2} + \cdots + P_{n}\] where \(P_{total}\) is the total pressure, and \(P_{1}, P_{2}, \cdots, P_{n}\) are the partial pressures of the individual gases.

Identify the given information

We are given the following information: - The mole ratio of nitrous oxide to oxygen is 1:1. - The total pressure in the cylinder is 2.50 atm.

Apply Dalton's Law to the given information

As we have a 1:1 ratio of gases, we can assume that both gases contribute equally to the total pressure. Therefore, the partial pressure of each gas will be half of the total pressure in the cylinder. We can express this as: \[P_{N2O} = P_{O2}\] Using the given total pressure of 2.50 atm, we can rewrite Dalton's Law as follows: \[2.50 atm = P_{N2O} + P_{O2}\]

Calculate the partial pressures of each gas

Since \(P_{N2O} = P_{O2}\), we can substitute one of these partial pressures into the equation, and solve for the partial pressure. \[2.50 atm = P_{N2O} + P_{N2O}\] Combine the like terms: \[2.50 atm = 2 P_{N2O}\] Now, divide by 2 to find the partial pressure of nitrous oxide: \[P_{N2O} = \frac{2.50 atm}{2} = 1.25 atm\] Since \(P_{N2O} = P_{O2}\), the partial pressure of oxygen is also 1.25 atm.

State the final answer

The partial pressure of nitrous oxide (N2O) and oxygen (O2) in the cylinder is 1.25 atm for each gas.

Key concepts

Partial Pressure Calculation

In the world of gases, partial pressure is like an individual gas's contribution to the overall pressure in a mixture. Dalton's Law of Partial Pressures helps us understand this concept beautifully.

According to Dalton's Law, the total pressure of a mixture of gases is the sum of the partial pressures of each gas present. This can be expressed as: \[P_{total} = P_{1} + P_{2} + \cdots + P_{n}\]where each \(P_n\) refers to the partial pressure of individual gases.

When you consider a mixture of nitrous oxide and oxygen with a 1:1 mole ratio, each gas influences the total pressure equally. Given a total pressure of 2.50 atm, each gas will exert a partial pressure of 1.25 atm. You arrive at this by dividing the total pressure by the number of gases in equal proportion: \[P_{N2O} = P_{O2} = \frac{2.50}{2} = 1.25 \text{ atm}\].

This straightforward division reflects the equal participation of each gas in producing the total pressure observed.

Mole Ratio of Gases

The mole ratio of gases gives us a snapshot of their presence relative to each other in a mixture. For instance, when nitrous oxide and oxygen are mixed in a 1:1 mole ratio, it means there's an equal amount of each gas present.

Understanding mole ratios is crucial because it simplifies the calculation of partial pressures. With a 1:1 ratio, both gases exert equal pressures if the total volume and temperature remain constant. This is because the number of moles — which directly correlates with pressure in a gas, given other conditions remain fixed — is the same for each gas.

In scenarios where the mole ratio isn't 1:1, those ratios become key in determining the percentage of the total pressure each gas contributes. Acting as a guide, the mole ratio simplifies understanding and calculating individual gas pressures in a mixture.

Non-reacting Gas Mixtures

Non-reacting gas mixtures are combinations of gases where the gases don't chemically interact with each other. Examples include air, which contains nitrogen, oxygen, argon, and carbon dioxide, or our nitrous oxide and oxygen mixture used in this problem.

In non-reacting gas mixtures, each gas maintains its individual property and behavior. Thus, the gases can be treated independently for calculations related to pressure, volume, and temperature.

The principle of Dalton’s Law leverages this behavior by treating the pressure contributed by each gas as if it were the only gas present. This means the total pressure of the gas mixture is merely the sum of each gas's partial pressure. Understanding non-reactive behavior allows us to predict and manipulate gas behaviors in various applications accurately.

Often, these applications range from breathing mixtures in deep-sea diving to anesthesia in medical settings, where controlling the gas mixtures' pressures ensures efficacy and safety.