Question

Benzaldehyde, a flavoring agent, is obtained by the dehydrogenation of benzyl alcohol. $$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{OH}(g) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CHO}(g)+\mathrm{H}_{2}(g)$$ \(K\) for the reaction at \(250^{\circ} \mathrm{C}\) is \(0.56\). If \(1.50 \mathrm{~g}\) of benzyl alcohol is placed in a 2.0-L flask and heated to \(250^{\circ} \mathrm{C}\), (a) what is the partial pressure of the benzaldehyde when equilibrium is established? (b) how many grams of benzyl alcohol remain at equilibrium?

Short answer

a) The partial pressure of benzaldehyde at equilibrium is approximately 0.74 atm. b) The mass of benzyl alcohol remaining at equilibrium is approximately 0.605 grams.

Step-by-step solution

Calculate the initial moles of benzyl alcohol

Convert the given mass of benzyl alcohol to moles using its molar mass (108.14 g/mol). $$\mathrm{Moles~of~benzyl~alcohol} = \frac{1.50~\mathrm{g}}{108.14~\mathrm{g/mol}} = 0.0139~\mathrm{mol}$$ Initially, no benzaldehyde or hydrogen are present, so their initial moles will be equal to 0.

Set up the ICE table

Using the initial moles calculated in step 1, create an ICE table to track the change in moles during the reaction. We will use 'x' to represent the change in moles. | Species | Initial moles | Change in moles | Equilibrium moles | |------------------|---------------|-----------------|-------------------| | Benzyl alcohol | 0.0139 | -x | 0.0139 - x | | Benzaldehyde | 0 | x | x | | Hydrogen | 0 | x | x |

Express the equilibrium constant in terms of moles and volume

Using the given value of K and the equilibrium expressions from the ICE table, we can write the equilibrium equation as follows: $$K = \frac{[\mathrm{C_{6}H_{5}CHO}][\mathrm{H_2}]}{[\mathrm{C_{6}H_{5}CH_2OH}]}$$ Convert the moles of each species to concentration by dividing them by volume (2 L): $$K = \frac{\frac{x}{2} \cdot \frac{x}{2}}{\frac{0.0139 - x}{2}}$$

Solve for the change in moles 'x'

Substitute the given value of K (0.56) into the equilibrium equation and solve for the change in moles 'x': $$0.56 = \frac{\frac{x^2}{4}}{\frac{0.0139 - x}{2}}$$ Solve the equation for x: $$x \approx 0.00832$$

Calculate the partial pressure of benzaldehyde

Convert the equilibrium moles of benzaldehyde to concentration by dividing by the volume (2 L): $$[\mathrm{C_{6}H_{5}CHO}] = \frac{x}{2} = \frac{0.00832~\mathrm{mol}}{2~\mathrm{L}} = 0.00416~\mathrm{M}$$ Now, using the ideal gas law, we can convert the molar concentration to partial pressure: $$P_{\mathrm{C_{6}H_{5}CHO}} = [\mathrm{C_{6}H_{5}CHO}] RT = (0.00416~\mathrm{M}) (0.0821~\mathrm{L \cdot atm / (mol \cdot K)}) (250+273.15)$$ $$P_{\mathrm{C_{6}H_{5}CHO}} \approx 0.74~\mathrm{atm}$$ The partial pressure of benzaldehyde at equilibrium is approximately 0.74 atm.

Determine the mass of benzyl alcohol at equilibrium

Use the equilibrium moles of benzyl alcohol from the ICE table and convert to mass using the molar mass: $$\mathrm{Mass~of~benzyl~alcohol~at~equilibrium} = (0.0139 - x) \cdot 108.14~\mathrm{g/mol}$$ $$\mathrm{Mass~of~benzyl~alcohol~at~equilibrium} = (0.0139 - 0.00832) \cdot 108.14~\mathrm{g/mol} \approx 0.605~\mathrm{g}$$ There are approximately 0.605 grams of benzyl alcohol remaining at equilibrium.

Key concepts

Understanding the ICE Table

In chemical equilibrium problems, an ICE table is a valuable tool. ICE stands for Initial, Change, and Equilibrium. It helps us keep track of the concentrations or moles of substances throughout a reaction. Initially, we list the moles present before the reaction begins. In this exercise, we start with 0.0139 moles of benzyl alcohol. Benzaldehyde and hydrogen begin with zero moles since they are products.
The change is represented by 'x', which stands for the moles of reactants converting into products. This reflects the stoichiometry of the reaction. As benzyl alcohol decomposes, its moles decrease by 'x', while benzaldehyde and hydrogen increase by 'x'.
Finally, the equilibrium line shows the moles at equilibrium: 0.0139 - x for benzyl alcohol and 'x' for the products. By setting up an ICE table, we can visualize and set up equations to find unknowns like the equilibrium constant or concentrations.

What is the Equilibrium Constant (K)?

The equilibrium constant, denoted as \(K\), is a crucial part of understanding chemical equilibrium. It relates the concentrations of reactants and products at equilibrium and reflects the extent of the reaction under specific conditions.
For the given reaction, benzyl alcohol converts to benzaldehyde and hydrogen. The formula for \(K\) is written as:

• \(K = \frac{[\text{C}_6\text{H}_5\text{CHO}][\text{H}_2]}{[\text{C}_6\text{H}_5\text{CH}_2\text{OH}]}\)

Here, each concentration is the equilibrium concentration.
The given \(K\) value is 0.56 at 250°C, indicating that at equilibrium the concentration of products is slightly less than that of the reactants. Solving for 'x' involves substituting known values and rearranging the equation. If \(K\) is much larger than 1, products are favored; if it's much smaller, reactants dominate.

Partial Pressure and Its Calculation

In the realm of gases, partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume alone. Calculating partial pressure involves knowing the concentration of a gas and applying the Ideal Gas Law.
For benzaldehyde, the concentration at equilibrium was found to be 0.00416 M. To convert this to partial pressure (\(P\)):

• Use the Ideal Gas Law: \(P = nRT\), where \(R\) is the gas constant (0.0821 L·atm/mol·K), and \(T\) is the temperature in Kelvin.

Inserting the values gives us a partial pressure of approximately 0.74 atm.
This concept is vital in systems where gases reach equilibrium, as pressures often determine reaction direction and extent, alongside concentrations. Partial pressures allow for better understanding and prediction of gaseous behavior in reactions.