Question

A gaseous mixture consists of $6.91\mathrm{g}$ of ${\mathrm{N}}_2,4.71\mathrm{g}$ of ${\mathrm{O}}_2,$ and $2.95\mathrm{g}$ of $\mathrm{He}.$ What volume does this mixture occupy at ${28}^{\circ }\mathrm{C}$ and 1.05 atm pressure?

Short answer

The volume of the gaseous mixture consisting of 6.91 g of N2, 4.71 g of O2, and 2.95 g of He at 28°C and 1.05 atm pressure is approximately 29.61 L.

Step-by-step solution

Mass to moles conversion

Since we have the mass of each gas, to convert it to moles, we will use the molar mass of the respective gases. The molar mass of N2 is 28 g/mol, the molar mass of O2 is 32 g/mol, and the molar mass of He is 4 g/mol. Moles of N2: ${\text{n}}_{{\text{N}}_2}=\frac{{\text{mass}}_{{\text{N}}_2}}{{\text{Molar\_Mass}}_{{\text{N}}_2}}=\frac{6.91\text{g}}{28\text{g/mol}}=0.2468\text{mol}$ Moles of O2: ${\text{n}}_{{\text{O}}_2}=\frac{{\text{mass}}_{{\text{O}}_2}}{{\text{Molar\_mass}}_{{\text{O}}_2}}=\frac{4.71\text{g}}{32\text{g/mol}}=0.1472\text{mol}$ Moles of He: ${\text{n}}_{\text{He}}=\frac{{\text{mass}}_{\text{He}}}{{\text{Molar\_mass}}_{\text{He}}}=\frac{2.95\text{g}}{4\text{g/mol}}=0.7375\text{mol}$

Find the total number of moles

Now that we have the moles for each gas, we need to find the total number of moles in the mixture: ${\text{n}}_{\text{total}}={\text{n}}_{{\text{N}}_2}+{\text{n}}_{{\text{O}}_2}+{\text{n}}_{\text{He}}=0.2468+0.1472+0.7375=1.1315\text{mol}$

Convert the temperature to Kelvin

The given temperature is in Celsius, and we need it in Kelvin for the ideal gas law equation. To convert it to Kelvin, we add 273.15 to the Celsius value: $\text{T (K)}={28}^{\circ }\text{C}+273.15=301.15\text{K}$

Solve for the volume using the ideal gas law

We now have all the necessary information to find the volume using the ideal gas law equation $PV=nRT$: $V=\frac{nRT}{P}$ Using R=0.0821 L.atm/mol.K and the given pressure P=1.05 atm, we have: $V=\frac{1.1315\text{mol}\times 0.0821\text{L.atm/mol.K}\times 301.15\text{K}}{1.05\text{atm}}=29.61\text{L}$ The volume of the gaseous mixture at 28°C and 1.05 atm pressure is approximately 29.61 L.

Key concepts

Mole Conversion

The concept of mole conversion is incredibly important in chemistry. It allows us to convert a given mass of a substance into the number of moles. This is beneficial because chemical reactions typically occur at the molecular level, where moles measure how many molecules are involved. To convert mass to moles, we use the formula:

• $extMoles=\frac{extmass(g)}{extmolarmass(g/mol)}$

In our exercise, we are dealing with three gases: nitrogen (N extsubscript{2}), oxygen (O extsubscript{2}), and helium (He). Each has a known mass, and their respective molar masses—28 g/mol for $ext{N}_2$, 32 g/mol for $ext{O}_2$, and 4 g/mol for $extHe$. By using the formula, we can convert each gas's mass into moles:

• For $ext{N}_2$, $6.91extg\to 0.2468extmol$
• For $ext{O}_2$, $4.71extg\to 0.1472extmol$
• For $extHe$, $2.95extg\to 0.7375extmol$

This mole conversion provides us the amount of each gas in molecules, which is crucial for further calculations in gas laws.

Molar Mass

Molar mass plays a key role in converting between grams and moles. It is defined as the mass of one mole of a given substance, measured in grams per mole (g/mol). Each element has a specific molar mass, which can often be found on the periodic table.
For example, in our exercise, nitrogen (N extsubscript{2}) has a molar mass of 28 g/mol because it consists of two nitrogen atoms, each with an atomic mass of about 14 g/mol. Similarly, oxygen (O extsubscript{2}) has a molar mass of 32 g/mol, as each oxygen atom has an atomic mass of approximately 16 g/mol. Helium (He), being a noble gas and monoatomic, has a molar mass of 4 g/mol.
Understanding and using molar mass is crucial for predicting how matter behaves during chemical reactions. With molar mass, we can transform mass into moles, which is essential for working with the Ideal Gas Law and other chemical equations.

Gas Mixture

A gas mixture involves the combination of different gases. Unlike liquids and solids, gases do not form layers but create homogenous mixtures. This means every part of a gas mixture has the same composition, so each gas occupies the same volume as if it were alone. It's crucial to calculate the total number of moles in gas mixtures to utilize gas law equations effectively.
In our case, we combined nitrogen, oxygen, and helium to form the mixture. Since gases mix fully and uniformly, we need the total number of moles for applying the Ideal Gas Law. We achieve this by calculating the sum from the individual moles derived from mole conversions:

• Total moles = moles of $ext{N}_2$ + moles of $ext{O}_2$ + moles of $extHe$
• Total moles = $0.2468+0.1472+0.7375=1.1315extmol$

Now, knowing the total moles is essential to predict the behavior of the gaseous mixture under specific conditions using the Ideal Gas Law.

Temperature Conversion to Kelvin

The Kelvin temperature scale is essential in physics and chemistry because it begins at absolute zero, the point where all molecular movement ceases. It's the standard temperature scale used in scientific calculations, particularly when dealing with equations like the Ideal Gas Law. To utilize this law correctly, we need temperatures in Kelvin.
Conversion from Celsius to Kelvin is straightforward and involves simply adding 273.15 to the Celsius temperature:

• $extTemperatureinKelvin=extTemperatureinCelsius+273.15$

In this exercise, we had a temperature of 28°C. Thus, by adding 273.15, we converted it to Kelvin:

• ${28}^{e}xtC+273.15=301.15extK$

Operating with the temperature in Kelvin ensures that our gas law calculations are accurate and reliable. This conversion is crucial whenever temperatures are included in calculations involving gases.