Problem 59 - Chemistry: The Central Science

Question

Hydrogen gas is produced when zinc reacts with sulfuric acid: $$ \mathrm{Zn}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{ZnSO}_{4}(a q)+\mathrm{H}_{2}(g) $$ If \(159 \mathrm{~mL}\) of wet \(\mathrm{H}_{2}\) is collected over water at \(24^{\circ} \mathrm{C}\) and a barometric pressure of 738 torr, how many grams of \(\mathrm{Zn}\) have been consumed? (The vapor pressure of water is tabulated in Appendix B.)

Step-by-Step Solution

Answer

In summary, given the volume of wet hydrogen gas collected, the temperature, and the barometric pressure, we calculated that 0.414 grams of Zinc have been consumed in the reaction with sulfuric acid to produce hydrogen gas.

Step 1: Calculate the moles of dry hydrogen gas

Using Ideal Gas Law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. First, we need to remove the vapor pressure of water at the given temperature since H2 was collected over water. From the given data: Temperature (T) = \(24^{\circ}C + 273.15\) = 297.15 K Volume of hydrogen gas (V) = 159 mL = 0.159 L Barometric pressure = 738 torr Vapor pressure of water at \(24^{\circ}C\) (from Appendix B) = 22.4 torr Now, we will find the partial pressure of hydrogen gas by subtracting the vapor pressure of water from the barometric pressure: Partial pressure of hydrogen gas = 738 torr - 22.4 torr = 715.6 torr Since the ideal gas constant R = 0.0821 L.atm/mol.K, we need to convert the pressure to atm: Pressure (P) = (715.6 torr) x (1 atm / 760 torr) = 0.9421 atm Now we can calculate the moles of dry hydrogen gas using the ideal gas law: \(n_{H_{2}} = \frac{PV}{RT} = \frac{(0.9421 \, atm)(0.159 \, L)}{(0.0821 \, L.atm/mol.K)(297.15 \, K)} = 0.00633 \, mol\)

Step 2: Determine the moles of Zinc consumed

From the balanced equation, the stoichiometric ratio between Zinc and Hydrogen gas is 1:1: \(Zn(s) + H_{2}SO_{4}(aq) \longrightarrow ZnSO_{4}(aq) + H_{2}(g)\) Since the ratio is 1:1, the moles of Zinc consumed are equal to the moles of hydrogen gas collected: \(n_{Zn} = n_{H_{2}} = 0.00633 \, mol\)

Step 3: Convert moles of Zinc consumed to grams

To calculate the mass of Zinc consumed, we need to multiply the moles of Zn consumed by its molar mass: Molar mass of Zn = 65.38 g/mol Mass of Zinc consumed = \(n_{Zn} \times \text{Molar mass of Zn} = (0.00633 \, mol)(65.38 \, g/mol) = 0.414 \, g\) Thus, 0.414 grams of Zinc have been consumed in the reaction.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and the amount of moles (n) of an ideal gas. This law is written as PV = nRT, where R is the ideal gas constant. The concept assumes that gases consist of particles that are in constant random motion and that these particles do not interact with one another except for brief elastic collisions.

To use the Ideal Gas Law, one must ensure that all units are consistent. Pressure is often measured in atmospheres (atm) or torr (where 1 atm = 760 torr), volume in liters (L), temperature in Kelvin (K), and the amount of substance in moles (mol). In practical scenarios, like the one involving hydrogen gas and zinc, the Ideal Gas Law allows us to calculate the moles of a gas when the pressure, volume, and temperature are known.

It is important to consider ambient conditions, such as humidity, when applying the Ideal Gas Law in experiments. For example, if a gas is collected over water, the vapor pressure of water must be subtracted from the total pressure to obtain the pressure of the dry gas alone. This correction ensures that the calculation accurately reflects the gas in question rather than a mixture with water vapor.

Chemical Reaction

A chemical reaction involves the transformation of one or more substances into new products. During a reaction, the bonds between atoms in the reactants are broken and new bonds form to create the products. In the provided exercise, the reaction between zinc (Zn) and sulfuric acid (H₂SO₄) to produce zinc sulfate (ZnSO₄) and hydrogen gas (H₂) is an example of a single displacement reaction, a common type of chemical reaction.

In such reactions, it is essential to balance the chemical equation. A balanced equation has an equal number of each type of atom on both sides, adhering to the Law of Conservation of Mass. Stoichiometry, the quantitative relationship between reactants and products in a chemical reaction, relies heavily on a balanced chemical equation to predict the amounts of materials consumed and produced.

In our scenario, the chemical reaction is already balanced and shows a 1:1 molar ratio between zinc and hydrogen gas. This information is crucial as it allows us to directly relate the moles of hydrogen gas produced to the moles of zinc consumed, thereby enabling the subsequent calculation of mass following the stoichiometric principles.

Molar Mass

Molar mass is the mass of one mole of a particular substance, expressed in grams per mole (g/mol). It is a fundamental concept in stoichiometry as it bridges the gap between the microscopic scale of atoms and molecules and the macroscopic world that we can measure and observe. The molar mass is numerically equal to the atomic or molecular weight of a substance, which can be found on the periodic table for individual elements and calculated for compounds by summing the atomic weights of their constituent elements.

For instance, the molar mass of zinc, used in the provided exercise, is 65.38 g/mol. Knowing the molar mass allowed the conversion from moles to grams, which is often the final step in a stoichiometry problem. To find the mass of a substance consumed or produced in a chemical reaction, you multiply the number of moles by the molar mass of the substance.

In the provided solution, after calculating the number of moles of zinc that reacted, the molar mass is used to find the mass of zinc consumed. This straightforward calculation is essential for translating quantitative data from the molecular level to practical measurements, a critical skill in experimental chemistry and other scientific endeavors.