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. To produce 240 mL of hydrogen gas at 30°C and 1.032 atm, 0.649 g of zinc must have reacted.

Step-by-step solution

Find the partial pressure of hydrogen gas

In order to find the partial pressure of hydrogen gas, we can use Dalton's law of partial pressures, which states that the total pressure of a gas mixture is equal to the sum of its partial pressures: $$P_{\text{total}} = P_{\text{H}_{2}} + P_{\text{H} _2 \text{O}}$$ We are given the total pressure (\(P_{\text{total}} = 1.032 \text{ atm}\)) and the vapor pressure of water at 30°C (\(P_{\text{H}_{2} \text{O}} = 32 \text{ torr}\)). We can convert \(P_{\text{H}_{2} \text{O}}\) to atm: $$P_{\text{H}_{2} \text{O}} = \frac{32 \text{ torr}}{760 \text{ torr/atm}} = 0.0421 \text{ atm}$$ Now we can find the partial pressure of hydrogen gas by using Dalton's law: $$P_{\text{H}_{2}} = P_{\text{total}} - P_{\text{H}_{2} \text{O}}$$

Calculate the partial pressure of hydrogen gas

Now we can substitute the pressure values into the equation and solve for the partial pressure of hydrogen gas: $$P_{\text{H}_{2}} = 1.032 \text{ atm} - 0.0421 \text{ atm} = 0.9899 \text{ atm}$$ The partial pressure of hydrogen gas is 0.9899 atm.

Find the number of moles of hydrogen gas

Now we need to find the number of moles of hydrogen gas to determine the amount of zinc reacted. We can use the Ideal Gas Law: $$PV = nRT$$ Rearranging the equation to find the number of moles (n): $$ n = \frac{PV}{RT}$$ We are given the volume (V) of hydrogen gas (240 mL = 0.240 L) and the temperature (T) is given in Celsius, but we need to convert it to Kelvin: $$T_K = 30°C + 273.15 = 303.15 \text{K}$$ The gas constant (R) in atm L/mol K is 0.0821, and we calculated the partial pressure of hydrogen gas (P) in step 2 as 0.9899 atm. Now we can plug in the values: $$n = \frac{(0.9899 \text{ atm})(0.240 \text{ L})}{(0.0821 \text{ atm L/mol K})(303.15 \text{K})}$$

Calculate the number of moles of hydrogen gas

Solve for the number of moles by substituting the values into the equation: $$n = \frac{(0.9899)(0.240)}{(0.0821)(303.15)} = 0.00994 \text{ mol}$$ The number of moles of hydrogen gas is 0.00994 mol.

Find the number of moles of zinc

Using the balanced equation given in the exercise, we can determine the number of moles of zinc required to produce the amount of hydrogen gas: $$\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ From the balanced equation, we can see that 1 mol of Zn reacts to produce 1 mol of H2. Therefore, the number of moles of zinc required is equal to the number of moles of hydrogen gas: $$\text{moles of Zn} = \text{moles of H}_{2} = 0.00994 \text{ mol}$$

Calculate the mass of zinc required

Finally, we can find the mass of zinc required to react by multiplying the number of moles of zinc by the molar mass of zinc: $$\text{mass of Zn} = \text{moles of Zn} \times \text{molar mass of Zn}$$ The molar mass of zinc is 65.38 g/mol. Therefore: $$\text{mass of Zn} = 0.00994 \text{ mol} \times 65.38 \frac{\text{g}}{\text{mol}}$$

Find the mass of zinc required

Multiply the number of moles by the molar mass to find the mass of zinc: $$\text{mass of Zn} = 0.00994 \times 65.38 = 0.649 \text{ g}$$ The mass of zinc required to produce this quantity of hydrogen gas is 0.649 g.

Key concepts

Hydrogen Gas Production

In the laboratory, producing small quantities of hydrogen gas is often achieved by reacting metallic zinc with aqueous hydrochloric acid. This reaction is described by the equation:

• Zn(s) + 2 HCl(aq) → ZnCl₂(aq) + H₂(g)

This equation illustrates that zinc reacts with hydrochloric acid to form zinc chloride and hydrogen gas. Hydrogen gas is typically collected by bubbling it through water. During this process, the hydrogen gas becomes saturated with water vapor. Understanding hydrogen gas production is crucial for calculating how much reactant, such as zinc, is needed to produce a given volume of hydrogen gas.

Ideal Gas Law Calculations

The Ideal Gas Law is a vital tool in chemistry for understanding the behavior of gases. It relates the pressure, volume, temperature, and number of moles of a gas using the equation:

• PV = nRT

Where:

• P is the pressure of the gas
• V is the volume
• n is the number of moles
• R is the gas constant (0.0821 atm L/mol K)
• T is the temperature in Kelvin

In the given problem, the Ideal Gas Law helps us determine how many moles of hydrogen gas are produced given a volume of 240 mL at a specific pressure and temperature. Knowing the number of moles allows us to find the mass of zinc needed through stoichiometry.

Vapor Pressure of Water

Vapor pressure is the pressure that the vapor of a liquid exerts when it is in equilibrium with its liquid phase. At a given temperature, water has a specific vapor pressure. In our example, at 30°C, the vapor pressure of water is 32 torr.

• This vapor pressure affects the total pressure of the gas mixture.

According to Dalton's law of partial pressures, the total pressure is the sum of the partial pressures of the gas components, including water vapor. Therefore, when measuring the total pressure during hydrogen gas collection, it is essential to account for the vapor pressure of water to find the actual pressure of the hydrogen gas itself. This step is critical for accurate calculations in chemical reactions involving gas collection over water.