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 Zn have been consumed? (The vapor pressure of water is tabulated in Appendix B.)

Short answer

0.418 grams of Zn have been consumed in the reaction.

Step-by-step solution

Convert volume and pressure of hydrogen gas to moles

First, adjust the pressure of hydrogen gas to exclude the vapor pressure of water, which is given as 24.6 torr at \(24^{\circ} C\). Total pressure of gases = Pressure of hydrogen gas + Vapor pressure of water Pressure of hydrogen gas = Total pressure of gases - Vapor pressure of water = \(738\: torr - 24.6\: torr\) = \(713.4\: torr\) Now, use the adjusted pressure of hydrogen gas, the volume of hydrogen gas, and the temperature to determine the moles of hydrogen gas using the Ideal Gas Law: \(PV=nRT\) Where: P = Pressure in atmospheres (\(101325\: Pa = 760\: torr\)) V = Volume in liters n = Moles of hydrogen gas R = Ideal gas constant (\(0.0821\: L\: atm/(mol\: K)\)) T = Temperature in Kelvin First, convert the given pressure and volume to the appropriate units: Pressure in atmospheres: \(\frac{713.4\: torr}{760\: torr/atm} = 0.9387\: atm\) Volume in liters: \(\frac{159\: mL}{1000\: mL/L} = 0.159\: L\) Temperature in Kelvin: \(24^{\circ}C + 273.15 = 297.15\: K\) Now, rearrange the Ideal Gas Law to solve for moles: \(n=\frac{PV}{RT}\) Substitute the values: \(n=\frac{(0.9387\: atm)(0.159\: L)}{(0.0821\: L\: atm/(mol\: K))(297.15\: K)}\) \(n=0.00639\: mol\: H_2\)

Determine moles of zinc consumed

Use the stoichiometry of the balanced equation to determine the moles of zinc consumed: \(\mathrm{Zn}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{ZnSO}_{4}(a q)+\mathrm{H}_{2}(g)\) According to the balanced equation, one mole of \(\mathrm{Zn}\) reacts with one mole of \(\mathrm{H_2}\). So, moles of zinc consumed = moles of hydrogen gas produced Moles of zinc consumed = \(0.00639\: mol\)

Convert moles of zinc to grams

Now, use the molar mass of zinc to convert moles of zinc to grams: Molar mass of \(\mathrm{Zn} = 65.38\: g/mol\) Grams of \(\mathrm{Zn}\) consumed = moles of \(\mathrm{Zn}\) consumed x molar mass of \(\mathrm{Zn}\) Grams of \(\mathrm{Zn}\) consumed = \(0.00639\: mol \times 65.38\: g/mol = 0.418\: g\) Therefore, 0.418 grams of Zn have been consumed in the reaction.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of an ideal gas. It is written as PV=nRT, where P stands for pressure, V denotes volume, n signifies moles of gas, R is the ideal gas constant, and T represents temperature in Kelvin.

This equation implies that, for a given quantity of gas, its pressure and volume are directly proportional to temperature and amount of gas. By manipulating this relationship, we can determine the amount of gas produced or consumed in a chemical reaction. For instance, given the volume, temperature, and pressure of hydrogen gas collected over water in the textbook problem, we can compute the moles of hydrogen gas. It's essential to adjust for the vapor pressure of water and convert measurements into standard units: volumes to liters, pressures to atmospheres, and temperatures to Kelvin, as done in the example.

When dealing with real gases, corrections may be necessary since real gases do not always behave ideally, especially under conditions of high pressure and low temperature. However, at relatively low pressures and high temperatures, the Ideal Gas Law provides a good approximation of the behavior of gases, making it an invaluable tool in stoichiometry and chemical calculations.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products. Understanding the stoichiometry, or the quantitative relationships between the amounts of reactants and products in a chemical reaction, is crucial. It allows chemists to predict how much reactant is needed to form a certain amount of product, or vice versa.

In a stoichiometric calculation, it's paramount to start with a balanced chemical equation. The balanced equation for the production of hydrogen gas in the example is \(\mathrm{Zn}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{ZnSO}_{4}(a q)+\mathrm{H}_{2}(g)\). This tells us that zinc and sulfuric acid react in a 1:1 molar ratio to produce zinc sulfate and hydrogen gas. With this key understanding, we can leverage the Ideal Gas Law to establish the moles of hydrogen gas collected, then use the stoichiometry of the reaction to determine the equivalent moles of zinc consumed.

It's important to recognize that various factors, such as temperature, pressure, catalyst presence, or the physical state of reactants, can influence the rate and completeness of a chemical reaction. When collecting gases over water, for instance, one must account for water's vapor pressure to ensure accurate measurements.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a bridge between the macroscopic world, which we can measure, and the microscopic world of atoms and molecules. Every element has a unique molar mass, easily found on the periodic table.

In the given problem, understanding molar mass allows us to convert moles of zinc to grams. The molar mass of zinc is \(65.38 \: g/mol\). Once we determine the moles of zinc reacting (established by stoichiometry from the moles of hydrogen gas formed), we simply multiply by zinc's molar mass to find the mass in grams. This conversion is a fundamental step in many chemistry problems.

It can be helpful to think of molar mass as a conversion factor that links the molecular scale to the laboratory scale. Accurate determination of molar mass is essential in calculating the quantities of reactants and products in a chemical reaction and in understanding the composition of compounds and elements.