Question

Ethene is converted to ethane by the reaction $$ \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \stackrel{\mathrm{Cindys}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{6}(g) $$ \(\mathrm{C}_{2} \mathrm{H}_{4}\) flows into a catalytic reactor at \(25.0 \mathrm{~atm}\) and \(300 .{ }^{\circ} \mathrm{C}\) with a flow rate of \(1000 .\) L/min. Hydrogen at \(25.0\) atm and \(300 .{ }^{\circ} \mathrm{C}\) flows into the reactor at a flow rate of \(1500 . \mathrm{L} / \mathrm{min}\). If \(15.0 \mathrm{~kg} \mathrm{C}_{2} \mathrm{H}_{6}\) is collected per minute, what is the percent yield of the reaction?

Short answer

The percent yield of the reaction between ethene and hydrogen gas to produce ethane is 92.66%. This value is calculated by determining the molar flow rate of the reactants, identifying the limiting reactant, calculating the theoretical and actual yield of ethane, and finally comparing the actual yield to the theoretical yield.

Step-by-step solution

Calculate molar flow rate of reactants

To determine the molar flow rate of the reactants, we first need to identify their molar masses and then use the given flow rates and conditions to calculate the moles entering the reactor per minute. For ethene (C2H4): \(M_{C2H4} = 2 \times 12.01 + 4 \times 1.01 = 28.05 \, g/mol\) For hydrogen (H2): \(M_{H2} = 2 \times 1.01 = 2.02 \, g/mol\) Next, using the ideal gas law \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the moles of the gas, \(R\) is the gas constant and \(T\) is temperature, we can obtain the number of moles per minute of the reactants. The given flow rates for ethene and hydrogen are 1000 L/min and 1500 L/min. The pressure is given as 25.0 atm, and the temperature is given as \(300^{\circ}C\) or 573.15 K. Using R = 0.0821 L atm/mol K: For ethene: \(n_{C2H4} = \frac{PV}{RT} = \frac{25.0 \times 1000}{0.0821 \times 573.15} = 538.27 \, mol/min\) For hydrogen: \(n_{H2} = \frac{PV}{RT} = \frac{25.0 \times 1500}{0.0821 \times 573.15} = 807.41 \, mol/min\)

Determine the limiting reactant and theoretical yield

The stoichiometry of the reaction is 1:1 for ethene and hydrogen to ethane. To determine the limiting reactant, we compare the molar flow rates of the reactants and identify the one with a lower ratio of its moles to its stoichiometric coefficient. Limiting reactant: \(min\left(\frac{n_{C2H4}}{1}, \frac{n_{H2}}{1}\right) = 538.27 \, mol/min\) From the stoichiometry of the reaction, we have 1 mol of ethene reacting with 1 mol of hydrogen to produce 1 mol of ethane. Thus, the theoretical yield of ethane based on the limiting reactant is equal to the molar flow rate of the limiting reactant. Theoretical yield of ethane: \(n_{C2H6, theory} = n_{C2H4, limit} = 538.27 \, mol/min\)

Calculate the actual yield

We are given that 15.0 kg of ethane is collected per minute. To find the actual yield, we need to convert this mass to moles by dividing by the molar mass of ethane. For ethane (C2H6): \(M_{C2H6} = 2 \times 12.01 + 6 \times 1.01 = 30.07 \, g/mol\) Actual yield of ethane: \(n_{C2H6, actual} = \frac{15.0 \times 10^3}{30.07} = 498.83 \, mol/min\)

Calculate the percent yield

Lastly, we will calculate the percent yield of the reaction by dividing the actual yield by the theoretical yield and multiplying by 100. Percent yield: \(\frac{n_{C2H6, actual}}{n_{C2H6, theory}} \times 100 = \frac{498.83}{538.27} \times 100 = 92.66 \% \) The percent yield of the reaction is 92.66%.

Key concepts

Stoichiometry

Stoichiometry is akin to a recipe for a chemical reaction. It involves determining the relative quantities of reactants and products involved in a reaction, ensuring the correct proportions to achieve the desired chemical transformation. Just as you would need the right amount of ingredients to bake a cake, in chemistry, stoichiometry ensures that reactants are mixed in the correct ratios according to the balanced chemical equation.

Let's think of it as following a cooking recipe. For each ethane molecule (C₂H₆) we want to make, we need one molecule of ethene (C₂H₄) and one molecule of hydrogen (H₂). The stoichiometry tells us these proportions are fixed, much like needing one egg for every cup of flour to make a batch of cookies.

Chemical Reaction

A chemical reaction involves the transformation of one set of chemical substances to another. Our scenario involves the chemical conversion of ethene to ethane. Imagine chemical reactions as social gatherings where certain guests (reactants) combine and convert into new groups (products) with different interactions. This particular party requires ethene and hydrogen to mingle in a very specific environment— facilitated by a catalyst and under high pressure and temperature—to become ethane.

Limiting Reactant

The limiting reactant in a chemical reaction is the ingredient that will be used up first, thus determining the amount of product that can be formed. In the kitchen, if you run out of eggs, you can't make more pancakes, regardless of how much milk and flour you have. Similarly, in the ethene-to-ethane reaction, the limiting reactant dictates the maximum amount of ethane produced. By calculating and comparing the molar flow rates, we can identify which reactant is the limiting 'egg' in our 'pancake' recipe.

Theoretical Yield

The theoretical yield is the maximum amount of product that can be produced in a chemical reaction based on the limiting reactant, assuming everything goes perfectly and no product is lost. Think of it as the number of pancakes you expect to make when you see the amount of ingredients you have available. The stoichiometry of our reaction provides a one-to-one relationship between reactants and product, which helps us calculate this ideal quantity of ethane that could be produced.

Actual Yield

In contrast to the ideal scenario, the actual yield is the amount of product really obtained from the chemical reaction. It's like actually making the pancakes and seeing how many you end up with—often fewer than expected due to batter sticking to the pan or being spilled. There are various reasons the actual yield is lower than theoretical yield, such as side reactions, incomplete reactions, or measurement errors.

Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical 'ideal' gas. It describes how gases behave under varying conditions of pressure, volume, and temperature, and it's useful to calculate the amount of substances involved in reactions that occur in the gas phase. Imagine blowing up a balloon—the size it reaches (volume) depends on how much you blow into it (amount of gas), how strongly you blow (pressure), and the warmth of the air or your breath (temperature). In our example, this law helps us figure out how many moles of gaseous ethene and hydrogen are entering the reactor.