Question

Small quantities of hydrogen gas can be prepared in the laboratory by the addition of aqueous hydrochloric acid to metallic zinc. $$\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ Typically, the hydrogen gas is bubbled through water for collection and becomes saturated with water vapor. Suppose 240\. mL of hydrogen gas is collected at \(30 .^{\circ} \mathrm{C}\) and has a total pressure of \(1.032\) atm by this process. What is the partial pressure of hydrogen gas in the sample? How many grams of zinc must have reacted to produce this quantity of hydrogen? (The vapor pressure of water is 32 torr at \(30^{\circ} \mathrm{C}\).)

Short answer

The partial pressure of hydrogen gas in the sample is 0.9899 atm, and 0.6538 grams of zinc must have reacted to produce this quantity of hydrogen gas.

Step-by-step solution

Find the partial pressure of water vapor

Given the vapor pressure of water at 30°C is 32 torr, now we need to convert it to atm in order to match the unit of the total pressure given in the problem. Conversion factor: 1 atm = 760 torr Partial pressure of water vapor = Vapor pressure of water at 30°C / 760 = 32 torr / 760 = 0.0421 atm

Calculate the partial pressure of hydrogen gas

Now we will use Dalton's Law to find the partial pressure of hydrogen gas. Total pressure = Partial pressure of hydrogen + Partial pressure of water vapor Partial pressure of hydrogen (P_H2) = Total pressure - Partial pressure of water vapor P_H2 = 1.032 atm - 0.0421 atm = 0.9899 atm

Determine moles of hydrogen gas collected

Next, we will use the ideal gas law to find the moles of hydrogen that are collected. Ideal gas law: PV = nRT, where P is pressure, V is volume, n is number of moles, R is the ideal gas constant, and T is temperature. Given the partial pressure of hydrogen gas (P_H2 = 0.9899 atm), volume (V = 240 mL or 0.240 L), and temperature (T = 30°C or 303.15 K), and the ideal gas constant (R = 0.0821 L_atm/mol_K). Rearrange the ideal gas law for n: n = PV / RT n_H2 = (0.9899 atm × 0.240 L) / (0.0821 L_atm/mol_K × 303.15 K) = 0.0100 mol We know that 0.0100 mol of hydrogen gas is collected.

Calculate moles of zinc reacted

From the balanced chemical equation, we know the stoichiometric ratio between zinc and hydrogen: $$\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ 1 mol Zn reacts with 1 mol H2 So the number of moles of zinc reacted is the same as the number of moles of hydrogen gas collected, which is 0.0100 mol.

Calculate grams of zinc reacted

Finally, we'll calculate the mass of zinc that must have reacted to produce the given quantity of hydrogen. Given the molar mass of zinc is 65.38 g/mol. Mass of zinc reacted = moles of zinc reacted × molar mass of zinc = 0.0100 mol × 65.38 g/mol = 0.6538 g Thus, 0.6538 grams of zinc must have reacted to produce this quantity of hydrogen gas.

Key concepts

Understanding Dalton's Law of Partial Pressures

Imagine you're at a party with a mix of different people talking all at once. The total noise is the sum of individual conversations, each contributing to the background sound. In a similar sense, Dalton's Law of Partial Pressures describes how a mixture of gases, like our group of partygoers, each contributes to the total pressure of the system.

In the scenario described in our exercise, hydrogen gas is collected with water vapor also present. Dalton's Law quantitatively relates the pressure of each individual gas (partial pressures) to the overall pressure within a container. The key equation we refer to is:
\[ P_{\text{total}} = P_1 + P_2 + P_3 + \cdots \]
Here, \( P_{\text{total}} \) is the total pressure, and \( P_1, P_2, P_3, \) etc., are the partial pressures of the individual gases in the mixture. When calculating the partial pressure of the hydrogen gas produced, it's necessary for students to subtract the known partial pressure of the water vapor (in atm from the vapor pressure in torr) from the total pressure to find the pressure that the hydrogen gas is exerting on its own.

Navigating the Ideal Gas Law Calculation

Stepping into the world of chemistry is akin to embarking on a space mission—precision is pivotal. To determine key variables like the number of moles of a gas, we employ the Ideal Gas Law, which is akin to our mission control formula for gases:
\[ PV = nRT \]
In this equation, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the ideal gas constant (0.0821 L_atm/mol_K), and \( T \) represents the temperature in Kelvin.

For the exercise, this law helps us chart the moles of hydrogen produced under given conditions of pressure, volume, and temperature. Knowing these parameters, we can rearrange the ideal gas law to solve for \( n \) (moles of gas), as demonstrated in the step-by-step solution. By understanding this equation, students can solve a wide variety of problems related to gas behavior under different conditions.

Demystifying Stoichiometry in Chemical Reactions

Chemical Ballet: A Dance of Proportions

Stoichiometry in chemical reactions is the mathematical relationship between reactants and products. It’s the choreography of a chemical dance, ensuring each dancer (reactant) pairs perfectly with their partner (product) according to the balanced chemical equation.

The stoichiometric calculations in our exercise are based on the reaction between zinc and hydrochloric acid to produce zinc chloride and hydrogen gas. It’s pivotal to understand that the coefficients in the balanced chemical equation tell us the exact ratio of moles of each reactant and product involved.

• 1 mole of zinc reacts with 2 moles of hydrochloric acid
• 1 mole of zinc produces 1 mole of hydrogen gas (since the coefficient in front of \( \text{H}_2 \) is 1).

By recognizing this 1:1 molar relationship between zinc and hydrogen, we confidently calculate the mass of zinc required to produce a measured volume of hydrogen gas. Students must always balance the equation first to ensure accurate stoichiometric calculations.