Question

A mixture of \(\mathrm{N}_{2}\) and Ne contains equal moles of each gas and has a total mass of \(10.0 \mathrm{~g}\). What is the density of this gas mixture at \(500 \mathrm{~K}\) and \(15.00 \mathrm{~atm}\) ? Assume ideal gas behavior.

Short answer

The density of the gas mixture is approximately 8.82 g/L.

Step-by-step solution

Identify Moles of Each Gas

Since the mixture contains equal moles of each gas, we start by denoting the number of moles of each gas as \( n \). Thus, the mixture contains \( n \) moles of \( \mathrm{N_2} \) and \( n \) moles of \( \mathrm{Ne} \), for a total of \( 2n \) moles.

Calculate Molecular Weights

The molar mass of \( \mathrm{N_2} \) is \( 28.0 \text{ g/mol} \) and for \( \mathrm{Ne} \) is \( 20.2 \text{ g/mol} \).

Set Up and Solve Mass Equation

The total mass of the gases is given as \( 10.0 \text{ g} \). Therefore, \( 28.0n + 20.2n = 10.0 \). Simplifying this equation gives \( 48.2n = 10.0 \). Solving for \( n \), we have \( n = \frac{10.0}{48.2} \approx 0.2075 \text{ moles} \).

Calculate Total Number of Moles

Since there are \( 2n \) moles in total, the total number of moles is \( 2 \times 0.2075 = 0.415 \text{ moles} \).

Calculate Volume using Ideal Gas Law

Using the Ideal Gas Law \( PV = nRT \), with \( P = 15.00 \text{ atm} \), \( n = 0.415 \text{ moles} \), \( R = 0.0821 \text{ L atm/mol K} \), and \( T = 500 \text{ K} \), we find \( V = \frac{nRT}{P} = \frac{0.415 imes 0.0821 imes 500}{15.00} \approx 1.134 \text{ L} \).

Calculate Density

Density \( \rho \) of the mixture is \( \frac{\text{mass}}{\text{volume}} \). Here, the total mass is \( 10.0 \text{ g} \) and the total volume is \( 1.134 \text{ L} \). Therefore, the density is \( \frac{10.0}{1.134} \approx 8.82 \text{ g/L} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry for understanding the behavior of gases under different conditions. It combines several gas laws into one formula:\[ PV = nRT \]Here:

• \(P\) stands for pressure (usually in atmospheres, atm).
• \(V\) is the volume that the gas occupies (commonly measured in liters, L).
• \(n\) represents the moles of the gas.
• \(R\) is the ideal gas constant \(0.0821 \,\text{L atm/mol K}\).
• \(T\) stands for temperature, where it must always be in Kelvin (K).

The equation helps predict how a gas will behave if any one of these variables changes, as long as you know the others. In this exercise, we used this law to find the volume of the gas mixture, by rearranging the formula to \( V = \frac{nRT}{P} \). Understanding this allows you to connect different properties of gases seamlessly.

Molar Mass

Molar Mass is an important concept that tells us the mass of one mole of a substance. It's expressed in grams per mole (g/mol) and helps to convert between grams of a substance and the number of moles. In our exercise, the molar masses of nitrogen (\(_{2}\)) and neon (Ne) were crucial.

• The molar mass of \(\text{N}_2\) is \(28.0 \,\text{g/mol}\).
• Whereas for \(\text{Ne}\), it’s \(20.2 \,\text{g/mol}\).

Molar mass is used in various calculations in chemistry, especially when involving reactions where you need to convert mass into moles or vice versa. This conversion is essential for determining the proportions of elements and compounds in chemical equations. In our solution process, we used these values to equate the total mass of the mixture to its total moles, facilitating accurate calculations.

Mole Concept

The Mole Concept is a cornerstone in chemistry, enabling you to quantify atoms and molecules using a unit called the mole. One mole equals \(6.022 \times 10^{23}\) entities—this number is Avogadro's number. In this exercise, we dealt with finding the mole quantity for the gas mixture.

• Moles provide a bridge between the atomic scale and the macroscopic quantities we measure in lab settings.
• The relationship is important because you can relate moles to mass, volume, and the number of particles.

In the problem, knowing we had equal moles of two gases allowed us to break down the calculations into manageable parts. When we calculated the total moles in the mixture, it included both gases, thereby setting up the necessary equations to follow through with further calculations like finding the gas volume using the ideal gas law.

Gas Mixtures

Gas Mixtures refer to the combination of two or more different gases. In this context, we examined a mixture consisting of nitrogen and neon. The properties of each gas in a mixture play a role in the overall properties of the gas mixture.

• When gases mix, each behaves as if it is alone in the container—this is a result of the ideal gas behavior assumption.
• The mixture's total volume and pressure can be considered the sum of each individual gas when viewed under the ideal conditions.
• These mixtures can have densities calculated by combining the individual properties as done in the problem.

Here, the mixture contained equal moles of \(\text{N}_2\) and \(\text{Ne}\). Knowing this, we calculated various properties like density using collective mass and volume. This exercise highlights the relevance of understanding gas mixtures to predict their behavior in different conditions.