Problem 55 - General Chemistry: Principles and Modern Applications

Question

What is the volume, in liters, occupied by a mixture of 15.2 \(\mathrm{g} \mathrm{Ne}(\mathrm{g})\) and \(34.8 \mathrm{g} \mathrm{Ar}(\mathrm{g})\) at 7.15 atm pressure and \(26.7^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The volume of the gas mixture is approximately 5.45 L

Step 1: Convert the masses of the gases to moles

The molar mass of Neon is 20.2 g/mol and that of Argon is 39.95 g/mol. So, the number of moles of Neon can be calculated as \(15.2 \, g \times \frac{1 mol}{20.2 \, g} = 0.75 \, mol\), and the number of moles of Argon can be calculated as \(34.8 \, g \times \frac{1 mol}{39.95 \, g} = 0.87 \, mol\).

Step 2: Sum the total moles and convert temperature

Add the moles of Neon and Argon to get the total moles of gases. So \(n = 0.75 \, mol + 0.87 \, mol = 1.62 \, mol\). Convert the temperature from Celsius to Kelvin by adding 273.15 (since \(K = ^\circ C + 273.15\)). So \(T = 26.7 + 273.15 = 299.85 \, K\).

Step 3: Calculate the volume using the Ideal Gas Law

Substitute the values you know into the equation PV = nRT and solve for V. The gas constant, R, when pressure is given in atm and volume in litres is \(0.0821 \, L.atm/K.mol\). So \(V = \frac{nRT}{P} = \frac{1.62 \, mol \times 0.0821 \, L.atm/K.mol \times 299.85 \, K}{7.15 \, atm}\)

Key Concepts

Molar Mass Calculation

Understanding how to calculate molar mass is key to solving many gas-related problems. **Molar mass** is the mass of one mole of a substance. To find the molar mass of an element, you look at its atomic mass on the periodic table. For compounds, you sum the atomic masses of all atoms in the formula.
For gases like Neon (Ne) and Argon (Ar), you directly use their atomic masses:

• The molar mass of Neon is 20.2 g/mol.
• The molar mass of Argon is 39.95 g/mol.

To convert grams to moles, you use the formula:
\[\text{Moles of substance} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}}\]For example, if you have 15.2 g of Neon, using its molar mass you calculate:

• \(15.2 \, g \times \frac{1 \, mol}{20.2 \, g} = 0.75 \, mol\)

This means you have 0.75 moles of Neon. You apply this same method to Argon.

Temperature Conversion to Kelvin

To solve any gas law problem, you need to convert the temperature to Kelvin because the Kelvin scale is absolute, starting at absolute zero. It aligns with the principles of the kinetic molecular theory.
**Converting Celsius to Kelvin** is straightforward. Use the formula:
\[K = ^\circ C + 273.15\]
Let's say your initial temperature is 26.7°C:

• Add 273.15 to 26.7, which equals 299.85 K.

Now, the temperature is properly converted, preparing you to insert it into the Ideal Gas Law equations. Remember, the Kelvin scale avoids negative values, simplifying calculations and ensuring consistent results in gas laws.

Gas Constant

The gas constant, often symbolized as \(R\), is vital for using the Ideal Gas Law. The Ideal Gas Law formula is given by:
\[PV = nRT\]
where:

• \(P\) = pressure in atm
• \(V\) = volume in liters
• \(n\) = number of moles
• \(R\) = gas constant
• \(T\) = temperature in Kelvin

In this equation, \(R\) has a specific value of \(0.0821 \, L.atm/K.mol\). This value is used when pressure is in atmospheres, volume in liters, and temperature in Kelvin. Plugging in these values correctly allows for the precise calculation of the volume of gas.
The beauty of \(R\) lies in its consistency. You can use it across various conditions for different gases, making it a universal tool in chemistry.