Question

Hydrogen gas is produced when zinc reacts with sulfuric acid: $$ \mathrm{Zn}(\mathrm{s})+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{ZnSO}_{4}(a q)+\mathrm{H}_{2}(g) $$ 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 Zn have been consumed? (The vapor pressure of water is tabulated in Appendix \(\mathbf{B} .\) )

Short answer

In this problem, we find the mass of Zn consumed by using the reaction stoichiometry, the ideal gas law, and vapor pressure of water. First, we calculate the partial pressure of hydrogen and the number of moles of hydrogen gas. Then, using the stoichiometry of the balanced chemical equation, we find the moles of Zn consumed. Finally, we convert the number of moles of Zn to grams, which gives us the result: approximately \(0.416\, \mathrm{g}\) of Zn have been consumed in the reaction.

Step-by-step solution

Write down the balanced chemical equation

The balanced chemical equation for the reaction between zinc and sulfuric acid is given as: \[ \mathrm{Zn}(\mathrm{s})+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{ZnSO}_{4}(a q)+\mathrm{H}_{2}(g) \]

Convert volume of hydrogen gas to moles using the ideal gas law

We will apply the ideal gas law formula (PV = nRT) to find the number of moles of hydrogen gas, where P is pressure, V is volume, n is the number of moles, R is the gas constant and T is the temperature. First, we need to correct the pressure and temperature into appropriate units: Temperature: \(24^{\circ} \mathrm{C} = (24+273.15) \mathrm{K} = 297.15 \mathrm{K}\) Pressure: \(738\, \mathrm{torr} = \dfrac{738}{760}\, \mathrm{atm} = 0.9711\, \mathrm{atm}\) Now, we need to correct the pressure for the vapor pressure of water at \(24^{\circ} \mathrm{C}\). From Appendix B (or using a standard reference), the vapor pressure of water at \(24^{\circ} \mathrm{C}\) is approximately \(22.4\, \mathrm{torr}\). Converting this to atm: \(22.4\, \mathrm{torr} = \dfrac{22.4}{760}\, \mathrm{atm} = 0.02947\, \mathrm{atm}\) Now we subtract this water vapor pressure from the measured pressure to find the partial pressure of hydrogen: \(P_\mathrm{H_2} = 0.9711 - 0.02947 = 0.9416\, \mathrm{atm}\) We can now solve for the number of moles (n) using the ideal gas law: \(PV = nRT \implies n = \dfrac{PV}{RT} \) Here, we use the gas constant \(R=0.0821\, \dfrac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}}\) The volume of hydrogen is given as \(159\, \mathrm{mL} = 0.159\, \mathrm{L}\) Substitute the values into the formula: \(n_\mathrm{H_2} = \dfrac{0.9416\, \mathrm{atm} \cdot 0.159\, \mathrm{L}}{0.0821\, \dfrac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}} \cdot 297.15 \mathrm{K}}\) Calculating the number of moles, we get: \(n_\mathrm{H_2} \approx 0.006365 \, \mathrm{mol}\)

Use stoichiometry to find moles of Zn consumed

From the balanced chemical equation, observe that one mole of Zn reacts completely with one mole of \(\mathrm{H_2}\): $\mathrm{Zn}\, : \, \mathrm{H_2} = 1 \, : \, 1$ Thus, the moles of Zn consumed are equal to the moles of \(\mathrm{H_2}\) produced: \(n_\mathrm{Zn} = n_\mathrm{H_2} = 0.006365\, \mathrm{mol}\)

Convert moles of Zn to grams

Using the molar mass of Zn, \(65.38\, \dfrac{\mathrm{g}}{\mathrm{mol}}\), we can calculate the mass of Zn consumed as follows: \(\mathrm{Mass\, of\, Zn} = n_\mathrm{Zn} \times M_\mathrm{Zn} = 0.006365\, \mathrm{mol} \times 65.38\, \dfrac{\mathrm{g}}{\mathrm{mol}}\) Calculating the mass of consumed Zn, we get: \(\mathrm{Mass\, of\, Zn} \approx 0.416\, \mathrm{g}\) So, about 0.416 grams of Zn have been consumed in the reaction.

Key concepts

Chemical Reactions

Understanding chemical reactions is fundamental to mastering stoichiometry in chemistry. A chemical reaction involves the transformation of one set of chemical substances to another. Governed by the laws of conservation of mass and energy, every chemical reaction is represented by a balanced chemical equation. This equation indicates how many atoms or molecules of reactants combine to form products.

In the given exercise, we examine the reaction between zinc and sulfuric acid, which produces zinc sulfate and hydrogen gas. The stoichiometry of the reaction is specified by the balanced equation, which indicates a one-to-one ratio between the reactants and the products they form. This ratio is crucial for stoichiometry calculations because it allows us to predict the amount of each substance involved in the reaction.

It’s important to note that stoichiometry does not only account for the number of particles but also their mass. The molar mass of each substance is essential for converting between moles and grams. By understanding these aspects of chemical reactions, students can solve problems regarding mass, volume, and particle number in a given reaction.

Ideal Gas Law

The ideal gas law is a pivotal concept when dealing with gases in chemical reactions. It combines several gas laws into the equation: \[PV = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. It's used to calculate the amount of gas either produced or consumed in a reaction under a given set of conditions.

In relation to the exercise, we're tasked with determining the amount of hydrogen gas produced. To do this, we need to ensure that we work with the correct units for temperature and pressure, and make the necessary corrections for variables like water vapor. Utilizing the ideal gas law helps us convert the volume of hydrogen gas collected to moles, which is a key step in finding the mass of zinc consumed.

It's essential to understand that this law applies to ideal gases, which are hypothetical gases that perfectly follow the kinetic molecular theory. While real gases do not perfectly obey the ideal gas law, it still provides a reasonable approximation for many gases under a variety of conditions.

Mole Concept

The mole concept is a cornerstone of chemistry and stoichiometry. It allows chemists to measure amounts of substances based on the number of particles rather than their mass. One mole is defined as exactly 6.02214076×10²³ elementary entities (Avogadro's number), whether they are atoms, ions, molecules, or electrons.

This concept is used in the exercise to relate the volume of hydrogen gas to the amount of zinc consumed. Since we are given the volume of hydrogen gas produced, we need to convert this volume to moles (using the ideal gas law) before we can find out how much zinc reacts. Given that reactions take place in fixed mole ratios, we use the mole concept to perform stoichiometric calculations to convert from moles of one substance to moles of another.

By using the molar mass of zinc, which relates the mass of a substance to its amount in moles, we can then convert the moles of consumed zinc to grams. Mastery of the mole concept is indispensable for solving a broad range of stoichiometry problems in chemistry.