To determine the volume of a 0.10 M hydrochloric acid (HCl) solution needed to produce 3.00 liters of hydrogen gas (H₂) according to the balanced chemical equation:
[tex]\[ 2 \text{HCl (aq)} + \text{Mg (s)} \rightarrow \text{MgCl}_2 \text{(aq)} + \text{H}_2 \text{(g)} \][/tex]
we can follow these steps:
### Step 1: Determine moles of H₂ gas produced
We start by using the ideal gas law to find the moles of H₂ gas produced. The ideal gas law is given by:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure (assumed to be 1 atm).
- [tex]\( V \)[/tex] is the volume of the gas (3.00 liters).
- [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L·atm/(K·mol)).
- [tex]\( T \)[/tex] is the temperature (assumed to be 298 K, which is 25°C).
Rearranging the equation to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Substituting the values:
[tex]\[ n_{H₂} = \frac{(1 \text{ atm}) \times (3.00 \text{ L})}{(0.0821 \text{ L·atm/(K·mol)}) \times (298 \text{ K})} \][/tex]
[tex]\[ n_{H₂} \approx 0.1226 \text{ mol} \][/tex]
### Step 2: Use the stoichiometry of the reaction
From the balanced chemical equation, 2 moles of HCl produce 1 mole of H₂. Therefore, the moles of HCl required to produce the calculated moles of H₂ is:
[tex]\[ n_{HCl} = 2 \times n_{H₂} \][/tex]
[tex]\[ n_{HCl} = 2 \times 0.1226 \text{ mol} \][/tex]
[tex]\[ n_{HCl} \approx 0.2452 \text{ mol} \][/tex]
### Step 3: Calculate the volume of HCl solution required
Finally, we use the molarity of the HCl solution to find the required volume. The molarity (M) is defined as moles of solute per liter of solution:
[tex]\[ \text{M} = \frac{\text{moles of solute}}{\text{liters of solution}} \][/tex]
Rearranging to solve for the volume of the solution:
[tex]\[ \text{Volume of HCl solution} = \frac{n_{HCl}}{\text{Molarity}} \][/tex]
Substituting the values:
[tex]\[ \text{Volume of HCl solution} = \frac{0.2452 \text{ mol}}{0.10 \text{ M}} \][/tex]
[tex]\[ \text{Volume of HCl solution} \approx 2.452 \text{ L} \][/tex]
So, to produce 3.00 liters of hydrogen gas, you need approximately 2.452 liters of a 0.10 M hydrochloric acid solution.
