Question

Consider the reaction: $$2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)$$ a. If 285.5 \(\mathrm{mL}\) of \(\mathrm{SO}_{2}\) reacts with 158.9 \(\mathrm{mL}\) of \(\mathrm{O}_{2}\) (both measured at 315 \(\mathrm{K}\) and 50.0 \(\mathrm{mm} \mathrm{Hg}\) , what is the limiting reactant and the theoretical yield of \(\mathrm{SO}_{3}\) ? b. If 187.2 \(\mathrm{mL}\) of \(\mathrm{SO}_{3}\) is collected (measured at 315 \(\mathrm{K}\) and 50.0 \(\mathrm{mmHg}\) ) what is the percent yield for the reaction?

Short answer

First calculate moles of SO2 and O2 using the ideal gas law. SO2 is the limiting reactant, producing a theoretical yield according to the stoichiometry. The percent yield is then found by comparing actual SO3 collected to the theoretical yield.

Step-by-step solution

Use the ideal gas law to calculate moles of SO2 and O2

Apply the ideal gas law PV = nRT to find the number of moles for both SO2 and O2. Use the given volumes, temperature (T = 315 K), pressure (P = 50.0 mmHg), and the universal gas constant (R = 62.36367 L mmHg /( K mol )) to calculate n (number of moles).

Convert pressure to atmospheres

1 atm = 760 mmHg. Convert the pressure of SO2 and O2 from mmHg to atm by dividing the given pressure by 760 mmHg/atm.

Determine the number of moles of SO2

Use the converted pressure in atm, the volume of SO2 in liters, the temperature in Kelvin, and R to solve for n (number of moles) for SO2. n = PV / (R x T).

Determine the number of moles of O2

Repeat the calculation from Step 3 for O2 using its respective volume to find the number of moles of O2.

Identify the limiting reactant

Use the stoichiometry of the balanced reaction, which shows a 2:1 mole ratio between SO2 and O2, to determine which reactant will be entirely consumed first, thus identifying the limiting reactant.

Calculate the theoretical yield of SO3

Using the limiting reactant's number of moles, use the stoichiometry to calculate the expected moles of SO3 produced (theoretical yield). Since the reaction ratio between SO2 and SO3 is 1:1, the moles of limiting reactant will equal the moles of SO3 produced.

Determine the percent yield

Finally, calculate the actual yield of SO3 using the ideal gas law with the given volume of SO3, then find the percent yield by dividing the actual yield by the theoretical yield and multiplying by 100 to get the percentage.

Key concepts

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the amounts of reactants and products in chemical reactions. It's based on the law of conservation of mass, which states that matter is neither created nor destroyed in chemical reactions. Therefore, we can relate the amounts of reactants and products through balanced chemical equations. A balanced equation shows the exact ratio of the number of moles of each substance involved.

For example, in the reaction between sulfur dioxide (SO₂) and oxygen (O₂), the balanced equation shows a 2:1 mole ratio. This means that two moles of SO₂ react with one mole of O₂ to produce two moles of sulfur trioxide (SO₃). The concept of the limiting reactant is fundamental in stoichiometry because it determines the maximum amount of product that can be made. The limiting reactant is the substance that is totally consumed first; once it's used up, the reaction cannot continue, and no additional product can be formed.

In a practical sense, to identify the limiting reactant, we compare the mole ratio of reactants used to the mole ratio indicated by the balanced equation. If the actual ratio does not match the equation's ratio, the reactant present in the lesser proportion is the limiting one.

Ideal Gas Law

The ideal gas law is an equation of state for a gas, where the state of an ideal gas is defined by its pressure (P), volume (V), temperature (T), and amount (n, in moles). The formula is expressed as PV = nRT, where R is the ideal gas constant. Despite the term 'ideal', this law provides an excellent approximation for the behavior of real gases under many conditions.

When using the ideal gas law, it's crucial to ensure that the units are consistent. For example, when the pressure is given in millimeters of mercury (mmHg), it should be converted to atmospheres (atm), since R is often given in terms of liters, atmospheres, moles, and Kelvin. Also, the volume should be in liters, temperature in Kelvin, and the amount in moles.

To apply this to a chemical reaction, you can use the ideal gas law to find the number of moles of a gaseous reactant or product, given the other variables. This is essential for solving problems where gas volumes are provided at a certain temperature and pressure and must be converted to moles to use in stoichiometric calculations.

Percent Yield

Percent yield is a measure of the efficiency of a chemical reaction, defined as the ratio of the actual yield to the theoretical yield, multiplied by 100%. The theoretical yield is the maximum amount of product that can be obtained from a reaction, assuming perfect conditions and complete conversion of reactants into products. It is calculated using the stoichiometry of the reaction based on the limiting reactant.

In a laboratory setting, the actual yield is the amount of product obtained from the reaction, which is often less than the theoretical yield due to real-world factors like incomplete reactions, side reactions, or loss of product during processing. To find the percent yield, take the actual yield (which can be measured or given in a problem statement), divide it by the theoretical yield (calculated from stoichiometry using the limiting reactant), and then multiply by 100% to express it as a percentage.

High percent yield indicates a more efficient reaction, which is a crucial concern in both academic studies and industrial chemical production. Understanding the factors that influence yield can help chemists optimize reactions for better efficiency and cost-effectiveness.