Question

Zinc metal reacts with hydrochloric acid to form hydrogen gas via the reaction \(\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \rightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)\) How many grams of zinc would be required to produce enough hydrogen gas to fill a 50.0-L cylinder to a pressure of \(10.0 \mathrm{~atm}\) at \(25.0^{\circ} \mathrm{C}\) ?

Short answer

To produce enough hydrogen gas to fill a 50.0 L cylinder to a pressure of 10.0 atm at 25.0 °C, we would require approximately 1331.36 grams of zinc metal.

Step-by-step solution

Determine moles of hydrogen gas using the Ideal Gas Law

The Ideal Gas Law is given by \(PV = nRT\), where: P - pressure, V - volume, n - number of moles, R - the Ideal Gas constant, T - temperature in Kelvin. First, we need to convert the given temperature to Kelvin by adding 273.15: \(25.0°C + 273.15 = 298.15 K\) Next, we rearrange the Ideal Gas Law equation to solve for the number of moles (n): \(n = \frac{PV}{RT}\) Substituting the given values (P = 10.0 atm, V = 50.0 L, T = 298.15 K, and using R = 0.0821 L·atm/mol·K as the gas constant): \(n = \frac{10.0 \, \text{atm} \cdot 50.0 \, \text{L}}{0.0821\, \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}} \cdot 298.15 \, \text{K}} \approx 20.37 \, \text{moles}\)

Convert moles of hydrogen gas to moles of zinc metal using stoichiometry

Using the given balanced chemical equation: \(\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \rightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)\) We have a 1:1 ratio of Zn to H₂, so the moles of Zn required will be equal to the moles of H₂ needed. 20.37 moles of H₂ = 20.37 moles of Zn.

Convert moles of zinc metal to grams

To find the mass of zinc metal required, we need to multiply the number of moles of zinc by its molar mass (65.38 g/mol): \(20.37 \, \text{moles Zn} \times 65.38 \, \frac{\text{g}}{\text{mol}} \approx 1331.36 \, \text{g}\) Thus, 1331.36 grams of zinc metal are required to produce enough hydrogen gas to fill a 50.0 L cylinder to a pressure of 10.0 atm at 25.0 °C.

Key concepts

Stoichiometry

Stoichiometry is a branch of chemistry that involves calculating the amounts of reactants and products in a chemical reaction. It's like the recipe of chemistry, ensuring that you mix the right proportions for the reaction to occur successfully.
In stoichiometry, we often rely on balanced chemical equations. These equations tell us the exact number of each molecule involved, such as in the reaction given:

• 1 molecule of zinc (Zn) reacts with 2 molecules of hydrochloric acid (HCl) to yield 1 molecule of zinc chloride (ZnCl₂) and 1 molecule of hydrogen gas (H₂).

The coefficients in these reactions indicate the ratio of moles of each substance. They help us translate between moles, grams, and even volume of gases, given the correct conditions.
For instance, in the problem, stoichiometry shows us that the moles of H₂ are equal to the moles of Zn because they both have a coefficient of 1, creating a neat 1:1 relationship.

Chemical Reactions

Chemical reactions represent the transformation of substances through the breaking and forming of bonds. In the reaction with zinc and hydrochloric acid, an oxidation-reduction process occurs.
Zinc metal (\(Zn(s)\)) gives up electrons to form zinc ions in solution, while hydrogen ions (\(H^+(aq)\)) in hydrochloric acid accept electrons to form hydrogen gas (\(H_2(g)\)).

• This process is mirrored in the balanced chemical equation:

\[\text{Zn}(s) + 2 \text{HCl}(aq) \rightarrow \text{ZnCl}_2(aq) + \text{H}_2(g)\]
Understanding these reactions means grasping not only the types of substances but also the process they undergo to transform. A balanced equation ensures conservation of mass, similar to how a balanced seesaw keeps both sides level.

Molar Mass

Molar mass is crucial in chemistry calculations, helping bridge between the microscopic scale of molecules and the macroscopic scale we deal with in labs. The molar mass is the mass of one mole of a substance.
It's derived from the atomic masses on the periodic table and is expressed in grams per mole (g/mol). For zinc, the molar mass is \(65.38 \, \text{g/mol}\).

• To find out how much zinc is needed, we multiplied the number of moles of zinc by its molar mass:
• \(20.37 \, \text{moles Zn} \times 65.38 \, \text{g/mol} \approx 1331.36 \, \text{g}\)

This gives us the actual mass of zinc required to generate the hydrogen gas needed in the problem.

Pressure and Temperature Conversions

Understanding and performing pressure and temperature conversions is necessary for solving problems involving gases, governed by the Ideal Gas Law. The law is expressed as \(PV = nRT\).
In this equation:

• P is the pressure, often in atmospheres (atm).
• V is the volume in liters (L).
• T is the temperature in Kelvin (K).
• R is the ideal gas constant (\(0.0821 \, \text{L} \cdot \text{atm/mol} \cdot \text{K}\)).

Temperature must always be in Kelvin for accurate calculations. To convert Celsius to Kelvin, simply add \(273.15\):
\(25.0°C + 273.15 = 298.15 \, \text{K}\)
By using these conversions correctly, you can determine the number of moles of gas needed with great precision, as illustrated in the zinc and hydrochloric acid reaction example.