Problem 61 - Chemistry The Central Science

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

Step-by-Step Solution

Answer

In summary, 0.428 grams of zinc have been consumed to produce 159 mL of wet hydrogen gas collected over water at 24°C and a barometric pressure of 738 torr.

Step 1: Determine the number of moles of hydrogen gas using the Ideal Gas Law

The ideal gas law is given by \(PV=nRT\), 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. We need pressure in atm, so we need to convert the pressure from torr to atm. Note that 1 atm = 760 torr. Also, we need to subtract the water's vapor pressure from the total pressure. Given: Barometric pressure (P_total) = 738 torr Vapor pressure of water (P_water) = From Appendix B (look up based on 24°C) = 22.4 torr (approximately) Volume (V) = 159 mL = 0.159 L Gas constant (R) = 0.08206 L atm K⁻¹ mol⁻¹ Temperature (T) = 24°C = 297 K Convert pressure from torr to atm: \(P_{H_2} = \frac{P_{total} - P_{water}}{760} = \frac{738 - 22.4}{760} = \frac{715.6}{760} = 0.9421\) atm Now we can use the ideal gas law to calculate the moles of hydrogen gas: \(PV = nRT \Rightarrow n = \frac{PV}{RT}\)

Step 2: Calculate the moles of hydrogen gas

Now, let's plug the values into the formula to find the moles of hydrogen gas: \(n_{H_2} = \frac{(0.9421)(0.159)}{(0.08206)(297)} = 0.00656\) moles of \(H_2\)

Step 3: Use stoichiometry to find moles of zinc

According to the balanced chemical equation, 1 mole of \(Zn\) reacts with 1 mole of sulfuric acid (\(H_2SO_4\)), producing 1 mole of zinc sulfate (\(ZnSO_4\)) and 1 mole of hydrogen gas (\(H_2\)). Therefore, the number of moles of \(Zn\) consumed will be the same as the number of moles of \(H_2\) produced: \(n_{Zn} = n_{H_2} = 0.00656\) moles of \(Zn\)

Step 4: Convert moles of zinc to grams

To find the mass of zinc consumed, multiply the number of moles of zinc by its molar mass. The molar mass of zinc is approximately 65.38 g/mol: Mass of Zn consumed = moles of \(Zn\) × molar mass of \(Zn\) Mass of Zn consumed = 0.00656 moles × 65.38 g/mol = 0.428 g Therefore, 0.428 grams of zinc have been consumed.

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is crucial for predicting the behavior of gases under different conditions. It's represented by the equation \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature in Kelvin.

When applying this law to a chemical reaction, it's essential to convert all units properly. For example, pressure must be in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K) to work with the standard value of the gas constant, \(0.08206 L atm K^{-1} mol^{-1}\). A common student pitfall is neglecting to convert temperature to Kelvin or pressure to atm, resulting in incorrect calculations. Always remember: Celsius to Kelvin requires adding 273.15, and to convert from torr to atm, divide the torr value by 760.

To put this into perspective, consider a balloon filled with helium. The volume of the balloon (\

The Reaction of Zinc with Sulfuric Acid

During the reaction between zinc and sulfuric acid, zinc metal is dissolved, and hydrogen gas is released as a byproduct. Represented by \(Zn(s) + H_2SO_4(aq) \rightarrow ZnSO_4(aq) + H_2(g)\), this is an example of a single displacement reaction. Understanding this process allows us to tackle stoichiometry challenges.

In stoichiometry, the law of conservation of mass states that matter is neither created nor destroyed. This principle implies that for every mole of zinc reacting, a mole of hydrogen gas is produced. In real-life applications, knowing this stoichiometric relationship can be essential in industrial processes where the purity of produced zinc sulfate or the amount of hydrogen gas release needs to be controlled. Students often confuse the reactants' and products' stoichiometric coefficients; always refer to the balanced equation for clarity.

Furthermore, when collecting gases over water, like hydrogen in this scenario, we must account for the vapor pressure exerted by water at the given temperature. This is subtracted from the overall pressure (in this example, barometric pressure) to find the pressure that the gas alone exerts. This is a detail that is sometimes overlooked but is critically important for accurate calculations.

The Mole Concept

The mole is a fundamental concept in chemistry that provides a bridge between the atomic and macroscopic worlds. One mole, which is Avogadro's number (\(6.022 \times 10^{23}\)), represents the quantity of any substance that contains as many elementary entities (like atoms or molecules) as there are atoms in 12 grams of carbon-12.

When working with the mole concept, always ensure that you're comfortable with converting between mass, moles, and the number of particles. For instance, the mass of one mole of a substance is equal to its molecular or atomic mass in grams. This conversion is vital when determining how much of a reactant is consumed or how much of a product is formed in a chemical reaction.

A common challenge for students is dealing with very small or very large numbers associated with mole calculations. To handle this, practice utilizing scientific notation and calculators effectively. Remember, understanding the mole concept is key to mastering stoichiometry, which in turn allows you to quantitatively analyze chemical reactions and processes.