Problem 49 - General Chemistry: Principles and Modern Applications

Question

A particular coal sample contains \(3.28 \%\) S by mass. When the coal is burned, the sulfur is converted to \(\mathrm{SO}_{2}(\mathrm{g}) .\) What volume of \(\mathrm{SO}_{2}(\mathrm{g}),\) measured at \(23^{\circ} \mathrm{C}\) and \(738 \mathrm{mm} \mathrm{Hg},\) is produced by burning \(1.2 \times 10^{6} \mathrm{kg}\) of this coal?

Step-by-Step Solution

Answer

The volume of SO2 produced by burning 1.2 x 10^6 kg of this coal under the given conditions is calculated using the steps above.

Step 1: Calculate the S mass in kg

First, we need to calculate the mass of sulfur in the coal. This is done by multiplying the total mass of the coal (1.2 x 10^6 kg) by the percentage of sulfur (3.28%) divided by 100. That is: \( S_{mass} = Coal_{mass} * S_{\%} / 100 \). Remember to convert the percentage to a decimal by dividing by 100.

Step 2: Convert S mass to moles of S

By using the molar mass of sulfur (S), we can convert the mass of sulfur calculated in step 1 to moles. The molar mass of S is approximately 32.066 g/mol but our mass is in kg so the molar mass becomes 32.066 kg/kmol. The conversion is done using the formula: \( S_{moles} = S_{mass} / S_{molar mass} \). This gives the amount of moles of sulfur.

Step 3: Convert moles of S to moles of SO2

Now we will convert moles of sulfur to moles of sulfur dioxide (SO2). According to the chemical equation of the burning process, one mole of S is converted to one mole of SO2. Therefore the amount of moles of sulfur dioxide is the same as the amount of moles of sulfur.

Step 4: Convert the given temperature and pressure

Before proceeding with the calculation of the volume, we need to convert the given temperature T (23°C) to an absolute temperature in Kelvin (K): \(T(K) = T(°C) + 273.15 \). Also, convert the pressure given in mmHg to atmospheres (atm) using the conversion factor \(1 atm = 760 mmHg\). The formula to use is: \( P(atm) = P(mmHg) / 760 \).

Step 5: Calculate the volume of SO2

Finally, we can use the ideal gas law equation to calculate the volume of the SO2. The ideal gas law is stated as \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the absolute temperature. For this problem, we know the value of n from step 3, P from the given pressure, and T from step 4. Also, R is 0.0821 L.atm/K.mol. Rearranging the ideal gas law equation to solve for V, we get: \( V = nRT / P \). Now, all that is left is to substitute the known values into the equation and calculate.

Key Concepts

Chemical Equations

Chemical equations are statements that depict the chemical reactions between substances. In a balanced chemical equation, the number of atoms for each element in the reactants should equal the number of atoms for those elements in the products. This upholds the law of conservation of mass, which states that mass cannot be created or destroyed in a chemical reaction.

In the given exercise, sulfur (S) in the coal is transformed into sulfur dioxide (SO₂) when burned. The simplified chemical equation for this transformation looks like this:

• S + O₂ → SO₂

Here, one mole of sulfur reacts with one mole of oxygen to produce one mole of sulfur dioxide. This equation indicates a one-to-one relationship between sulfur and sulfur dioxide, which is useful for stoichiometry calculations.

Whenever you are dealing with chemical reaction problems like this, ensure you first write down the balanced chemical equation, as it will guide the conversion of quantities between different substances involved.

Ideal Gas Law

The ideal gas law provides a relationship between the four variables involved in a gas sample: pressure (P), volume (V), number of moles (n), and temperature (T). It is expressed in the formula:

• PV = nRT

Where R is the ideal gas constant, usually 0.0821 L.atm/K.mol.

In this exercise, you're tasked with finding the volume of sulfur dioxide gas produced, knowing other quantities. By rearranging the ideal gas law formula, you can solve for volume (V):

• V = nRT / P

Here's how you use it:
- **Pressure (P)** needs to be converted to atmospheres if not already. - **Temperature (T)** should be in Kelvin, so convert it by adding 273.15 to the Celsius temperature. - **Number of moles (n)** was calculated in previous steps, based on the mass of sulfur.

By substituting these known values in the equation, the volume of the gas can be found easily. Understanding the ideal gas law is crucial for predicting how gases will behave under various conditions.

Mole Concept

The mole is a fundamental concept in chemistry that bridges atomic and macroscopic worlds. A mole is an amount of substance containing Avogadro's number ( 6.022×10²³) of particles (atoms, molecules, ions, etc.). The concept allows chemists to count entities in a practical way and is especially useful in stoichiometry.

In this exercise, you start with the mass of sulfur present in coal. To find how much sulfur you have in terms of moles, you'll use the molar mass of sulfur (approximately 32.066 g/mol). The formula to convert mass to moles is:

• Moles = Mass / Molar Mass

By calculating moles, you help set the stage for further stoichiometric calculations, converting moles of sulfur to moles of sulfur dioxide based on the chemical equation.

Mastering the mole concept is vital for understanding how substances interact, as it allows scaling from microscale reactions to real-world quantities.