Problem 110 - Chemistry The Central Science In Si Units

Question

(a) A sample of hydrogen gas is generated in a closed container by reacting \(1.750 \mathrm{~g}\) of zinc metal with \(50.0 \mathrm{~mL}\) of \(1.00 \mathrm{M}\) hydrochloric acid. Write the balanced equation for the reaction, and calculate the number of moles of hydrogen formed, assuming that the reaction is complete. (b) The volume over the solution in the container is 150 \(\mathrm{mL}\). Calculate the partial pressure of the hydrogen gas in this volume at \(25^{\circ} \mathrm{C}\), ignoring any solubility of the gas in the solution. (c) The Henry's law constant for hydrogen in water at \(25^{\circ} \mathrm{C}\) is \(7.7 \times 10^{-6} \mathrm{~mol} / \mathrm{m}^{3}-\mathrm{Pa}\). Estimate the number of moles of hydrogen gas that remain dissolved in the solution. What fraction of the gas molecules in the system is dissolved in the solution? Was it reasonable to ignore any dissolved hydrogen in part (b)?

Step-by-Step Solution

Answer

(a) 0.0250 mol H₂ produced, (b) Partial Pressure = 4.09 atm, (c) Dissolved fraction = 0.64%; assumption in Part B was reasonable.

Step 1: Write the Balanced Chemical Equation

The reaction between zinc metal (Zn) and hydrochloric acid (HCl) is a single replacement reaction, producing zinc chloride (ZnCl₂) and hydrogen gas (H₂).The balanced chemical equation for this reaction is:\[ \text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2 \]

Step 2: Calculate Moles of Reactants

First, calculate the moles of zinc. The molar mass of zinc (Zn) is 65.38 g/mol. Using the formula:\[ \text{Moles of Zn} = \frac{1.750 \text{ g}}{65.38 \text{ g/mol}} \approx 0.02676 \text{ mol} \]Now calculate the moles of HCl. The volume of HCl solution is 50.0 mL, which is 0.0500 L. With a concentration of 1.00 M:\[ \text{Moles of HCl} = 1.00 \frac{\text{mol}}{\text{L}} \times 0.0500 \text{ L} = 0.0500 \text{ mol} \]

Step 3: Determine Limiting Reagent

The reaction requires 2 moles of HCl for every mole of Zn. With 0.02676 mol of Zn, \[ \text{Required moles of HCl} = 2 \times 0.02676 = 0.05352 \text{ mol} \]Since only 0.0500 mol of HCl is available, HCl is the limiting reagent.

Step 4: Calculate Moles of Hydrogen Gas Formed

Since HCl is the limiting reagent, use it to calculate the moles of hydrogen gas (H₂) formed. According to the balanced reaction:\[ 2 \text{ mol HCl} \rightarrow 1 \text{ mol H₂} \]So, 0.0500 mol HCl will produce:\[ \frac{0.0500}{2} = 0.0250 \text{ mol H₂} \]

Step 5: Calculate Partial Pressure of Hydrogen Gas

Using the ideal gas law to calculate the partial pressure of hydrogen gas over the solution. The volume over the solution is 150 mL, which is 0.150 L. At 25°C (298 K), utilizing the ideal gas law:\[ PV = nRT \]Rearrange to solve for pressure (P):\[ P = \frac{nRT}{V} \]Where:- \(n = 0.0250\) mol- \(R = 0.0821 \frac{\text{L atm}}{\text{K mol}}\)- \(T = 298\) K- \(V = 0.150\) LCalculating the pressure,\[ P = \frac{0.0250 \times 0.0821 \times 298}{0.150} \approx 4.09 \text{ atm} \]

Step 6: Estimate Dissolved Hydrogen in the Solution

Using Henry's law, calculate the amount of hydrogen gas dissolved.Henry's law states:\[ C = kP \]Where:- \(C\) is the concentration of dissolved gas (mol/m³)- \(k = 7.7 \times 10^{-6}\) mol/(m³ Pa)- \(P = 4.09\) atm Convert atm to Pa:\[ P_{Pa} = 4.09 \times 101325 \approx 414414.25 \text{ Pa} \]Calculate the concentration:\[ C = 7.7 \times 10^{-6} \times 414414.25 = 3.19 \times 10^{-3} \text{ mol/m}^3 \]Given the volume of solution is 0.0500 L (0.0500 m³), the moles of dissolved gas:\[ \text{Moles of dissolved H₂} = 3.19 \times 10^{-3} \times 0.050 \approx 1.60 \times 10^{-4} \text{ mol} \]

Step 7: Calculate Fraction of Dissolved Gas

Calculate the fraction of gas that is dissolved in the solution.\[ \text{Fraction of dissolved gas} = \frac{1.60 \times 10^{-4}}{0.0250} \approx 0.0064 \]This means that approximately 0.64% of the hydrogen gas is dissolved.

Step 8: Assess Assumption for Part B

Since the dissolved hydrogen is only about 0.64% of the total, ignoring the dissolved hydrogen when calculating partial pressure in Part B was reasonable.

Key Concepts

Limiting Reagent

In chemistry, reactions often involve more than one reactant, and the amounts of these reactants determine how much product is formed. However, the reactant that is completely used up first limits the extent of the reaction.This is known as the limiting reagent.

For example, in the reaction between zinc and hydrochloric acid to produce hydrogen gas, there are two reactants: zinc (\(\text{Zn}\)) and hydrochloric acid (\(\text{HCl}\)).

• To determine the limiting reagent, we calculate moles of each reactant. Here, 1.750 g of zinc equates to about 0.02676 moles.
• Meanwhile, there are 0.0500 moles of hydrochloric acid present.
  
• The balanced chemical equation for this reaction is \(\text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2\), implying that 2 moles of HCl are required for each mole of Zn consumed.

Given these amounts, hydrogen chloride is the limiting reagent since less than 0.05352 moles (2 times the moles of Zn) are present. This limits the extent of the reaction despite zinc being available in excess.

Ideal Gas Law

The Ideal Gas Law gives us a good approximation of how gases behave under various conditions of temperature and pressure. It is represented by the formula:\(PV = nRT\), where:

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) represents moles of gas,
• \(R\) is the ideal gas constant (commonly 0.0821 L atm/K mol),
• and \(T\) is the temperature in Kelvin.
  

For hydrogen gas inside the closed container in our exercise, we use this law to calculate the pressure of the gas once the reaction reaches completion.First, convert the reacting conditions:

• The produced hydrogen gas fills a volume of 0.150 L, and it exists at a temperature of \(25^{\circ} \text{C}\) or 298 K.
  
• Knowing \(n = 0.0250\) mol, we apply the ideal gas law: \(P = \frac{nRT}{V}\).
  

This calculation gives us a partial pressure of approximately 4.09 atm.
Understanding how gases like hydrogen behave under these conditions helps predict outcomes in various chemical processes.

Henry's Law

Henry's Law is essential when determining how gases dissolve in liquids, specific to pressure. The law states that the concentration of dissolved gas (\(C\)) is proportional to its pressure above the liquid, represented by:\(C = kP\).

• In this equation, \(k\) is the Henry's Law constant, differing depending on the gas and the liquid.
• In our exercise, \(k\) is \(7.7 \times 10^{-6}\) mol/m³-Pa for hydrogen in water.

To estimate hydrogen's solubility in water:

• First, convert the partial pressure from atm to Pa, giving \(P = 414414.25\) Pa.
• This converts to a concentration \(C = 3.19 \times 10^{-3}\) mol/m³, assuming 0.0500 L of water (or 0.0500 m³).
• The dissolved moles of hydrogen are about \(1.60 \times 10^{-4}\) mol.
  

Finally, evaluating the proportion of dissolved hydrogen in the system, only about 0.64% is dissolved, affirming the negligible impact of dissolved hydrogen on the overall system, especially in pressure calculations.
Henry's Law clarifies the usually small contribution of gas solubility under specific reaction conditions.