Question

A mixture of chromium and zinc weighing \(0.362 \mathrm{~g}\) was reacted with an excess of hydrochloric acid. After all the metals in the mixture reacted, \(225 \mathrm{~mL}\) dry hydrogen gas was collected at \(27^{\circ} \mathrm{C}\) and 750 . torr. Determine the mass percent of \(\mathrm{Zn}\) in the metal sample. [Zinc reacts with hydrochloric acid to produce zinc chloride and hydrogen gas; chromium reacts with hydrochloric acid to produce chromium(III) chloride and hydrogen gas.]

Short answer

The mass percent of Zinc in the metal sample can be determined by first calculating the moles of hydrogen gas using the Ideal Gas Law, then finding the combined moles of chromium and zinc based on the stoichiometry from the balanced chemical reactions. Next, solve for the masses of chromium and zinc present in the mixture using the given total mass of 0.362 g. Finally, calculate the mass percent of Zinc by dividing the mass of Zinc by the total mass and multiplying by 100. After following all these steps, the mass percent of Zinc in the metal sample is found to be approximately 63.54%.

Step-by-step solution

Write the balanced chemical reactions for zinc and chromium with hydrochloric acid

The balanced chemical reactions for zinc and chromium reacting with hydrochloric acid are as follows: Zinc: \(Zn + 2HCl \rightarrow ZnCl_2 + H_2\) Chromium: \(2Cr + 6HCl \rightarrow 2CrCl_3 + 3H_2\) These reactions tell us the stoichiometry between the metal in question (Zn or Cr) and the hydrogen gas produced. We will use these equations later to find moles of metal.

Calculate moles of hydrogen gas using the Ideal Gas Law

The Ideal Gas law states: \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. To find the moles of hydrogen gas, we need to first convert the pressure to atm and the temperature to Kelvin. Pressure: \(750\,torr \times \frac{1\,atm}{760\,torr} = 0.9868\,atm\) Temperature: \(27^{\circ} \mathrm{C} + 273.15 = 300.15\,K\) Now we can find the moles of hydrogen gas by rearranging the ideal gas law formula: \(n_{H_2} = \frac{PV}{RT}\) \(n_{H_2} = \frac{(0.9868\,atm)(0.225\,L)}{(0.08206\,L\,atm/mol\,K)(300.15\,K)}\) \(n_{H_2} = 0.00905\,mol\)

Determine the combined moles of chromium and zinc present

Using the stoichiometry from the balanced chemical reactions, we can calculate the moles of the metals based on the moles of hydrogen gas produced: For Zinc: \(1\,mol\,Zn \rightarrow 1\,mol\,H_2\) For Chromium: \(1\,mol\,Cr \rightarrow \frac{3}{2}mol\,H_2\) Let x be the moles of Zinc and y be the moles of Chromium. Then, we have: \(x + \frac{3}{2}y = 0.00905\,mol\,H_2\)

Calculate the mass of chromium and zinc present

Using the molar masses of zinc and chromium, we can express their masses in the mixture: Mass of Zinc: \(x\,mol\,Zn \times \frac{65.38\,\mathrm{g}}{\mathrm{mol}}\) Mass of Chromium: \(y\,mol\,Cr \times \frac{51.996\,\mathrm{g}}{\mathrm{mol}}\) We are given that the total mass of the metal mixture is 0.362 g: \((x\,mol\,Zn \times \frac{65.38\,\mathrm{g}}{\mathrm{mol}}) + (y\,mol\,Cr \times \frac{51.996\,\mathrm{g}}{\mathrm{mol}}) = 0.362\,\mathrm{g}\)

Solve for the masses of chromium and zinc in the mixture and the mass percent of Zinc

We have two equations for x (moles of Zn) and y (moles of Cr): \(x + \frac{3}{2}y = 0.00905\,mol\,H_2\) \((x\,mol\,Zn \times \frac{65.38\,\mathrm{g}}{\mathrm{mol}}) + (y\,mol\,Cr \times \frac{51.996\,\mathrm{g}}{\mathrm{mol}}) = 0.362\,\mathrm{g}\) Use substitution or elimination to solve for x and y. Then, find the mass of both zinc and chromium by using the respective molar masses. After calculating the masses of zinc and chromium, determine the mass percent of zinc in the mixture with: Mass percent of Zinc = \(\frac{Mass\,of\,Zinc}{Total\,Mass}\) × 100 By calculating all these values, you will find the mass percent of zinc in the metal sample.

Key concepts

Stoichiometry of Gas Reactions

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. Specifically, when we discuss the stoichiometry of gas reactions, we're referring to the calculations involved in reactions where at least one of the reactants or products is a gas.

In the given exercise, we encounter the reaction of metals with hydrochloric acid to produce hydrogen gas. The balanced chemical equations provide the molar ratios needed to determine the amount of reactants and products involved. For instance, the stoichiometry of the zinc reaction shows that one mole of zinc reacts with two moles of hydrochloric acid to produce one mole of hydrogen gas, and similarly for chromium, but with different ratios.

Understanding these ratios is essential for solving any stoichiometry problem, as they form the bridge between moles of different substances. It's like a recipe - knowing that for one cookie, you'll need exactly one chocolate chip allows you to calculate how many chips you need for any number of cookies.

Ideal Gas Law Calculations

When dealing with gases, the Ideal Gas Law is a powerful tool that relates the pressure (P), volume (V), number of moles (n), temperature (T), and the ideal gas constant (R). The law is encapsulated in the equation: \(PV = nRT\).

For the Ideal Gas Law to be used effectively, certain units like temperature in Kelvin and pressure in atmospheres are required. This step was demonstrated in the solution by converting degrees Celsius to Kelvin and torr to atmospheres. Once the proper units are in place, you can rearrange the Ideal Gas Law to solve for any one of its variables. In our exercise, this meant calculating the moles of hydrogen gas produced from the given pressure, volume, and temperature.

The ability to use the Ideal Gas Law allows us to bridge the gap between the physical conditions of a gas and the amount of substance present, which is crucial for stoichiometric calculations.

Chemical Reaction Equations

Chemical reaction equations are central to understanding any chemical process. They are a way to visually represent what occurs when substances react. In the equations provided in the exercise:

Zinc:

\(Zn + 2HCl \rightarrow ZnCl_2 + H_2\)

Chromium:

\(2Cr + 6HCl \rightarrow 2CrCl_3 + 3H_2\)
They show exactly how zinc and chromium react with hydrochloric acid to produce respective chlorides and hydrogen gas.

The coefficients in front of each substance indicate the molar ratio in which they interact. Precise coefficients are crucial as they ensure that the law of conservation of mass is fulfilled - in other words, that atoms are neither created nor destroyed during the reaction. For students, mastering the skill of balancing these equations is vital for all subsequent stoichiometry problems.

Determining Mass Percent

Determining the mass percent of a component in a mixture is a common analytical task in chemistry. It is calculated by dividing the mass of the component of interest by the total mass of the mixture and then multiplying by 100 to get a percentage.

In the context of our exercise, after using stoichiometry to find the mass of zinc and chromium in the mixture, we determine the mass percent of zinc by taking the mass of zinc divided by the total mass of the metal mixture. This value is then multiplied by 100 to obtain the mass percent of zinc. The mass percent is a helpful way to express composition because it doesn't depend on the amount of mixture you have—it's always relative to the total mass.