Question

What volume of \(\mathrm{H}_{2}(\mathrm{g})\) at \(25^{\circ} \mathrm{C}\) and \(752 \mathrm{mmHg}\) is required to hydrogenate oleic acid, \(\mathrm{C}_{17} \mathrm{H}_{33} \mathrm{COOH}(1)\) to produce one mole of stearic acid, \(\mathrm{C}_{17} \mathrm{H}_{35} \mathrm{COOH}(\mathrm{s}) ?\) Assume reaction (22.52) proceeds with a 95\% yield.

Short answer

The required volume of \(\mathrm{H}_{2}(\mathrm{g})\) is approximately 26.3 L.

Step-by-step solution

Write the balanced chemical equation

Firstly, the balanced chemical equation for the hydrogenation of oleic acid is required to be written. This can be written as: \(\mathrm{C}_{17}\mathrm{H}_{33} \mathrm{COOH} + \mathrm{H}_{2} \rightarrow \mathrm{C}_{17} \mathrm{H}_{35} \mathrm{COOH}\). Here it can be seen that one mole of \(\mathrm{H}_{2}(\mathrm{g})\) is required to produce one mole of stearic acid.

Adjust mole quantity for yield

The reaction proceeds with a 95 \% yield, meaning that only 95 \% of the theoretical amount is actually used. This implies that for one mole of stearic acid to be produced, \(1 \, mole/\ 0.95 = 1.05 \, moles\) of \(\mathrm{H}_{2}(\mathrm{g})\) are actually required.

Calculate volume using Ideal Gas Law

With the use of the Ideal Gas Law \(PV = nRT\), the volume of the \(\mathrm{H}_{2}(\mathrm{g})\) can be calculated. In this case, \(P = 752 \, mmHg = 0.99 \, atm\), \(T = 25^{\circ}C = 298 \, K\), \(n = 1.05 \, moles\), and \(R = 0.0821 \, L \cdot atm/(K \cdot mole)\). Plugging these values into the formula results in: \(V = nRT/P = 1.05 \, mole \cdot 0.0821 \, L \cdot atm/(K \cdot mole) \cdot 298 \, K / 0.99 \, atm = 26.3 \, L\).

Conversion to significant digits

Giving the answer in three significant figures, as the given data all have three significant figures, the volume is around 26.3 L.

Key concepts

Hydrogenation

Hydrogenation is a chemical reaction that involves the addition of hydrogen (H₂) to another compound. In the context of this exercise, oleic acid undergoes hydrogenation to become stearic acid.

• Hydrogenation is often used to convert unsaturated fats to saturated fats.
• This is achieved by breaking the double bonds in the carbon chain and adding hydrogen atoms, making the molecule more stable.

In this reaction, each molecule of oleic acid reacts with a single molecule of hydrogen gas.
This transforms the unsaturated oleic acid into fully saturated stearic acid, resulting in a product that is solid at room temperature.
The hydrogenation process is crucial in many industrial applications, including the production of margarine and other hydrogenated fats.

Chemical Equation Balancing

Balancing chemical equations is essential in understanding the stoichiometry of a reaction. It ensures that the law of conservation of mass is upheld. In this exercise, the balanced chemical equation is given as:\[\mathrm{C}_{17}\mathrm{H}_{33} \mathrm{COOH} + \mathrm{H}_{2} \rightarrow \mathrm{C}_{17} \mathrm{H}_{35} \mathrm{COOH}\]

• This equation shows one molecule of hydrogen reacting with one molecule of oleic acid to produce stearic acid.
• Balancing the equation ensures each molecule's atoms align, confirming that no atoms are lost or added outside the reaction itself.

Balancing a chemical equation is a fundamental skill in chemistry, as it lays the groundwork for calculating how much of each reactant is necessary or how much product will be formed.

Reaction Yield

The concept of reaction yield is important in determining how efficient a reaction is.

• In theory, a reaction would convert all reactants to products.
• However, in practice, the "actual yield" is often less than the "theoretical yield" due to incomplete reactions or side reactions.

For this problem, the reaction has a 95% yield.To calculate the actual amount of hydrogen needed, consider:\[\text{Moles required} = \frac{1 \text{ mole of stearic acid}}{0.95} = 1.05 \text{ moles of } \mathrm{H}_{2}\]This implies that while theoretically, only one mole of hydrogen is needed per mole of oleic acid, the actual requirement is 1.05 moles due to the 95% efficiency.
Understanding yield allows chemists to adjust reactant amounts and plan for imperfect conditions in the lab or industry.

Volume Calculation

Calculating the volume of a gas under specific conditions can be done using the Ideal Gas Law: \(PV = nRT\).

• This equation relates the pressure (P), volume (V), moles of a gas (n), the gas constant (R), and temperature (T).
• It's instrumental when dealing with gases in chemical reactions, providing insight into properties like volume and pressure under different conditions.

In the exercise, the calculation uses:- Pressure: 0.99 atm (converted from 752 mmHg)- Temperature: 298 K (converted from 25°C)- Moles of Hydrogen: 1.05- Ideal gas constant: 0.0821 L·atm/(K·mole)By substituting these values into the Ideal Gas Law:\[V = \frac{nRT}{P} = \frac{1.05 \times 0.0821 \times 298}{0.99} \approx 26.3 \text{ L}\]This calculation shows that approximately 26.3 liters of hydrogen gas are needed under the given conditions. Understanding volume calculations are crucial for laboratory setups and industrial processes involving gases.