Problem 152 - Chemistry An Atoms First Approach

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 of 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.]

Step-by-Step Solution

Answer

The mass percent of zinc in the metal sample is 38.7%.

Step 1: Calculate the moles of hydrogen gas produced

To calculate the moles of hydrogen gas produced, we can use the ideal gas equation: \(PV = nRT\) Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. First, we need to convert the pressure to atmospheres and the temperature to Kelvin: Pressure (atm) = \(\frac{750 \text{ torr}}{760\text{ torr/atm}}\) = 0.9868 atm Temperature (K) = 27°C + 273.15 = 300.15 K Volume (L) = \(\frac{225 \text{ mL}}{1000\text{ mL/L}}\) = 0.225 L Now we can calculate the number of moles of hydrogen gas produced: \(n = \frac{PV}{RT}\) \(n_{\text{H}_{2}} = \frac{(0.9868\text{ atm})(0.225\text{ L})}{(0.0821\text{ L atm/mol K})(300.15\text{ K})}\) \(n_{\text{H}_{2}} = 0.00905 \text{ mol}\)

Step 2: Calculate the moles of zinc and chromium reacted

Now we will use the stoichiometry of the reactions to determine the moles of zinc and chromium reacted. The balanced chemical equations are: \(Zn + 2HCl \rightarrow ZnCl_{2} + \text{H}_{2}\) \(2Cr + 6HCl \rightarrow 2CrCl_{3} + 3\text{H}_{2}\) From these equations, we see that 1 mole of zinc will produce 1 mole of hydrogen gas, and 1 mole of chromium will produce 1.5 moles of hydrogen gas. We can use these ratios to set up a system of equations: \(n_{Zn} + n_{Cr} = n_{\text{H}_{2}}\) \(\frac{1}{2} n_{Cr} = n_{Zn}\) We can substitute the value of \(n_{\text{H}_{2}}\) into the first equation: \(n_{Zn} + n_{Cr} = 0.00905 \text{ mol}\) And substitute the value of \(\frac{1}{2} n_{Cr}\) for \(n_{Zn}\) in the second equation: \(\frac{1}{2} n_{Cr} + n_{Cr} = 0.00905 \text{ mol}\)

Step 3: Determine the mass percent of zinc

First, we will find the moles of chromium: \(1.5 n_{Cr} = 0.00905 \text{ mol}\) \(n_{Cr} = 0.00603 \text{ mol}\) Now, we can use the value of \(\frac{1}{2} n_{Cr}\) to find the moles of zinc: \(n_{Zn} = \frac{1}{2} n_{Cr} = \frac{1}{2} (0.00603 \text{ mol}) = 0.00302 \text{ mol}\) Next, we can calculate the mass of zinc and chromium in the sample: Mass of zinc = \(m_{Zn} = n_{Zn} \times M_{Zn} = 0.00302 \text{ mol} \times 65.38\text{ g/mol} = 0.197 \text{ g}\) Mass of chromium = \(m_{Cr} = n_{Cr} \times M_{Cr} = 0.00603 \text{ mol} \times 51.996\text{ g/mol} = 0.313 \text{ g}\) Finally, we can calculate the mass percent of zinc in the sample: Mass percent of zinc = \(\frac{m_{Zn}}{m_{Zn} + m_{Cr}} \times 100\% = \frac{0.197\text{ g}}{0.197\text{ g} + 0.313\text{ g}} \times 100\% = 38.7\%\) So, the mass percent of zinc in the metal sample is 38.7%.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a crucial formula in chemistry used to relate the properties of gases: pressure, volume, temperature, and amount of substance in moles. The formula is given by \( PV = nRT \), where \( P \) stands for pressure (usually in atmospheres), \( V \) for volume (in liters), \( n \) for the number of moles, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin.

To apply this formula properly, it's essential to ensure that each variable's unit matches the required type. For example, pressure should be in atmospheres rather than torr, and volume should be in liters rather than milliliters. Similarly, temperature must be converted to Kelvin by adding 273.15 to the Celsius degree.

• Convert pressure using \( 750 \text{ torr} = 0.9868 \text{ atm} \).
• Convert temperature with \( 27^{\circ} \text{C} + 273.15 = 300.15 \text{ K} \).
• Convert volume from milliliters to liters: \( 225 \text{ mL} = 0.225 \text{ L} \).

This preparation lets us calculate the moles of gases produced, like hydrogen in our problem.

Stoichiometry

Stoichiometry is the area of chemistry concerned with measuring and calculating the amounts of reactants and products in chemical reactions. It relies heavily on balanced chemical equations, which indicate the quantitative relationship between reactants and products.

The key to stoichiometry is understanding the ratios given by the balanced equations, which in turn allow you to determine how much of each reactant is consumed and how much product is formed. This can be particularly useful in reactions involving gases, where the ideal gas law further helps to determine the moles of the substances involved.

• Form the ratios from balanced chemical reactions.
• Translate moles of products back to moles of reactants.

In the exercise, knowing that 1 mole of Zn reacts to produce 1 mole of \( \text{H}_2 \), and 1 mole of Cr produces 1.5 moles of \( \text{H}_2 \), we construct a system of equations to explore the reactants further.

Balanced Chemical Equations

Balanced chemical equations are equations where the quantities of all reactive and product species are equal on both sides. This is a fundamental aspect because atoms are conserved in reactions.

In the problem, we have two equations:

• \( \text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2 \)
• \( 2\text{Cr} + 6\text{HCl} \rightarrow 2\text{CrCl}_3 + 3\text{H}_2 \)

These equations show a precise relationship between the amounts of reactants and products. For instance, the first equation tells us that the reaction of zinc with hydrochloric acid directly produces hydrogen gas in a 1:1 ratio. Contrast that with chromium's 2:3 ratio in the second equation.

This involves setting up equations where the individual gas production results from each metal's reaction, which allows for mole calculations to find each metal's proportion in the original mixture.

Mole Calculation

Mole calculation is converting between mass and number of particles using Avogadro's number, but it also relates directly to chemical reactions via stoichiometry.

Given the moles of \( \text{H}_2 \) from the ideal gas calculations, the mole calculations for Zn and Cr extend using stoichiometry. For zinc, which forms 1 mole hydrogen per mole of metal, and chromium, producing 1.5 moles hydrogen per 2 molecules of metal, we first determine mole values.

• Calculated \( n_{\text{H}_2} = 0.00905 \text{ mol} \).
• Setup the mole relations: \( n_{\text{Zn}} + n_{\text{Cr}} = n_{\text{H}_2} \).
• Solve system of equations from stoichiometric relations.

This system lets us find how many moles of each metal participated. Finally, convert these moles to masses using the molar mass of each metal, which contributes pivotal data to find the mass percentage of zinc.