Question

In a flask of volume \(V\) litres, \(0.2\) mol of axygen, \(0.4\) mol of nitrogen, \(0.1\) mol of ammonia and \(0.3 \mathrm{~mol}\) of helium are endosed at \(27^{\circ} \mathrm{C}\). If the total pressure exerted by these non-reacting gases is one atmosphere, the partial pressure exerted by nitrogen is (a) \(0.1\) atmosphere (d) \(0.4\) atmosphere (c) \(0.3\) atmosphere

Short answer

The partial pressure exerted by nitrogen is 0.4 atmosphere.

Step-by-step solution

Understanding 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 by the mixture is equal to the sum of the partial pressures of individual gases. The partial pressure of each gas is directly proportional to its mole fraction in the container.

Calculate the total moles of gas

To find the mole fraction of nitrogen, you must first determine the total moles of gas in the container. Add the moles of all the given gases together: \[0.2 \, \text{mol O}_2 + 0.4 \, \text{mol N}_2 + 0.1 \, \text{mol NH}_3 + 0.3 \, \text{mol He} = 1.0 \, \text{mol total gas}\].

Determine the mole fraction of nitrogen

The mole fraction of nitrogen ((x_{N_2}) is the number of moles of nitrogen divided by the total moles of gas: \[x_{N_2} = \frac{0.4 \, \text{mol}}{1.0 \, \text{mol total gas}} = 0.4\].

Calculate the partial pressure of nitrogen

Using Dalton's Law, the partial pressure of nitrogen ((P_{N_2})) is found by multiplying the mole fraction of nitrogen by the total pressure: \[P_{N_2} = x_{N_2} \times P_{\text{total}} = 0.4 \times 1 \, \text{atmosphere} = 0.4 \, \text{atmosphere}\].

Key concepts

Partial Pressure Calculation

Understanding how to calculate partial pressure is fundamental when dealing with gas mixtures. It's based on Dalton's Law of Partial Pressures, which states the total pressure exerted by a mixture of non-reacting gases is equal to the sum of their individual pressures. Each gas contributes a partial pressure proportional to its mole fraction. To compute a gas's partial pressure, first determine the total pressure exerted by the gas mixture. Next, calculate the mole fraction for each gas, which is the ratio of the number of moles of a particular gas to the total number of moles of all gases present. The partial pressure for any gas is then the product of its mole staffraction and the total pressure.

For example, if a container holds several gases at a total pressure of 1 atmosphere, and you need to find the partial pressure of nitrogen (), which is present at 0.4 moles in a 1.0 mole mixture, you would use the formula:
. This results in a partial pressure for nitrogen of 0.4 atmospheres, indicating how each gas's presence influences the overall pressure of the gas mixture.

Mole Fraction Concept

The mole fraction concept is a way of expressing the concentration of a component in a mixture. Calculating a substance's mole fraction involves dividing the number of moles of that substance by the total number of moles of all substances present. It's a unitless number that represents the proportion of a component in the mixture.

Mole fraction is often represented by the Greek letter . In the context of gases, if you're given several gases in a container with known amounts in moles, you find the total moles and then divide the moles of each individual gas by this total to obtain their respective mole fractions. For instance, if nitrogen is 0.4 moles and the total is 1.0 moles, the mole fraction of nitrogen () is -- it's a simple yet powerful dimensionless ratio that's crucial in many applications including calculating partial pressures and determining composition in chemical reactions and processes.

Behavior of Non-reacting Gas Mixtures

Non-reacting gas mixtures, such as those mentioned in the exercise, exhibit behavior that can be predicted by Dalton's Law. Since the gases do not chemically interact, each gas within the mixture exerts pressure as if it were the only gas present. This independent behavior allows us to use the concept of partial pressures to describe their collective behavior within a container.
Behavior characteristics of non-reacting gas mixtures include:

The gases do not influence each other's physical properties. Each gas behaves according to its own identity, following the ideal gas law individually.

The total pressure of the mixture is the sum of each gas's partial pressure, making it straightforward to calculate when you know the mole fraction and the total pressure.

The presence of one gas does not affect the volume occupied by another; they distribute evenly throughout the entire volume of their container.

These properties ensure that, in the absence of chemical reactions, gas mixtures can be approached in a simplistic, yet accurate manner, using Dalton's Law for calculations of pressure and other related properties.