Question

A solution is prepared by adding \(50.0 \mathrm{~mL}\) concentrated hydrochloric acid and \(20.0 \mathrm{~mL}\) concentrated nitric acid to 300 \(\mathrm{mL}\) water. More water is added until the final volume is \(1.00\) L. Calculate \(\left[\mathrm{H}^{+}\right],\left[\mathrm{OH}^{-}\right]\), and the \(\mathrm{pH}\) for this solution. [Hint: Concentrated HCl is \(38 \%\) HCl (by mass) and has a density of \(1.19 \mathrm{~g} / \mathrm{mL} ;\) concentrated \(\mathrm{HNO}_{3}\) is \(70 . \% \mathrm{HNO}_{3}\) (by mass) and has a density of \(1.42 \mathrm{~g} / \mathrm{mL}\).]

Short answer

The final concentrations and pH of the solution are: \([\mathrm{H}^{+}] = 0.907 \, \text{M}\), \([\mathrm{OH}^{-}] = 1.10 \times 10^{-14} \, \text{M}\), and \( \text{pH} = 0.042\).

Step-by-step solution

Calculate the moles of HCl and HNO3 in the solution

First, we need to find the moles of HCl and HNO3 in the solution. We are given the volume of each concentrated acid, their percentage by mass, and their density. We can use this information to find the mass of each acid in the solution and then calculate the moles: Moles of HCl = (volume of HCl × density of HCl × mass percentage of HCl) / molecular weight of HCl Moles of HNO3 = (volume of HNO3 × density of HNO3 × mass percentage of HNO3) / molecular weight of HNO3 Molecular weight of HCl = 36.5 g/mol Molecular weight of HNO3 = 63 g/mol Moles of HCl = (50.0 mL × 1.19 g/mL × 0.38) / 36.5 g/mol = 0.610 mol Moles of HNO3 = (20.0 mL × 1.42 g/mL × 0.7) / 63 g/mol = 0.297 mol

Calculate the total hydrogen ion concentration

Now that we have the number of moles of HCl and HNO3, we can calculate the total moles of hydrogen ions in the solution, as both acids ionize completely: Total moles of H+ ions = moles of HCl + moles of HNO3 Total moles of H+ ions = 0.610 mol + 0.297 mol = 0.907 mol The final volume of the solution is 1.00 L, so we can now find the hydrogen ion concentration: \[ [\mathrm{H}^+] = \frac{0.907 \, \text{mol}}{1.00 \, \text{L}} = 0.907 \, \text{M} \]

Calculate the hydroxide ion concentration

We can use the ion product constant of water (Kw) to find the hydroxide ion concentration: \[K_{w} = [\mathrm{H}^{+}] [\mathrm{OH}^{-}]\] The ion product constant of water is \(K_{w} = 1.0 \times 10^{-14} \, \text{M}^{2}\) \[ [\mathrm{OH}^{-}] = \frac{K_{w}}{[\mathrm{H}^{+}]} \] \[ [\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14} \, \text{M}^{2}}{0.907 \, \text{M}} = 1.10 \times 10^{-14} \, \text{M} \]

Calculate the pH

Now that we have the hydrogen ion concentration, we can calculate the pH of the solution using the following formula: \[ \text{pH} = -\log_{10} [\mathrm{H}^{+}] \] pH = -log10(0.907 M) = 0.042 The final concentrations and pH of the solution are as follows: \[\ [\mathrm{H}^{+}] = 0.907 \, \text{M} \] \[\ [\mathrm{OH}^{-}] = 1.10 \times 10^{-14} \, \text{M} \] \[\ \text{pH} = 0.042 \]

Key concepts

Acid Concentration

When we talk about the acid concentration in a solution, we're referring to the amount of acid present in a given volume of solution. It's often expressed as molarity (M), which is the number of moles of acid per liter of solution. This is crucial for calculating the pH of the solution because the more concentrated the acid, the higher the amount of hydrogen ions \( \mathrm{H}^+ \) it will produce upon dissociation.

For example, in the step by step solution provided, the concentration of hydrochloric acid (HCl) and nitric acid (\( \mathrm{HNO}_3 \)) are determined by using their densities and mass percentages to find the mass of the acids in the solution. This mass is then converted to moles, which ultimately leads to the calculation of the concentration when the final volume of the solution is considered. For our exercise, calculations showed that there were 0.610 moles of HCl and 0.297 moles of \( \mathrm{HNO}_3 \).

Tips for Better Understanding:

• Always ensure that the volume unit matches the desired concentration units; mL must be converted to L for molarity.
• Remember that the percentage (mass/volume) indicates how much of the substance is present in a certain amount of solvent, and it's key to finding the mole quantity.
• Knowing the molar masses of the acids is essential for the conversion from mass to moles.

Hydrogen Ion Concentration

The hydrogen ion concentration, denoted as \( [\mathrm{H}^+] \), is a direct measure of the acidity of a solution. As acids dissociate, they release hydrogen ions into the solution; the more hydrogen ions present, the more acidic the solution is, which also affects its pH value. For strong acids like HCl and \( \mathrm{HNO}_3 \), which dissociate completely in water, the hydrogen ion concentration can be equated to the acid concentration.

In our exercise, we added the moles of HCl and \( \mathrm{HNO}_3 \) to find the total moles of hydrogen ions (0.907 moles), and since the final volume of the solution was 1.00 L, the hydrogen ion concentration was 0.907 M. This high concentration indicates a very acidic solution.

Key Points to Remember:

• Hydrogen ion concentration is central to calculations of pH, which puts a logarithmic scale on the acidity of a solution.
• For strong acids, the mole-to-mole ratio between the acid and the hydrogen ions is typically 1:1.
• Ensure you use the correct final volume of the solution to calculate the hydrogen ion concentration after dilution.

Hydroxide Ion Concentration

Conversely, the hydroxide ion concentration \( [\mathrm{OH}^-] \) refers to the concentration of hydroxide ions in a solution, and it's an indicator of the solution's basicity. The hydroxide ion concentration is related to the hydrogen ion concentration through the water dissociation constant (\( K_w \) at 25°C is \( 1.0 \times 10^{-14} \) M²). For every dissociated hydrogen ion in an aqueous solution, a hydroxide ion is also present because of water's self-ionization.

In the provided solution, once we found the hydrogen ion concentration, we used the ion product constant of water to calculate the hydroxide ion concentration: \( [\mathrm{OH}^-] = \frac{K_w}{[\mathrm{H}^+]} \), which in our case yielded a very low value of \( 1.10 \times 10^{-14} \) M due to the high concentration of hydrogen ions.

Understand the Balance:

• The product of hydrogen ion and hydroxide ion concentrations will always equal the water dissociation constant (\( K_w \) for water).
• As the solution becomes more acidic, the hydroxide ion concentration decreases, which is a reciprocal relationship represented by the constant \( K_w \).
• In neutral water at 25°C, \( [\mathrm{H}^+] \) and \( [\mathrm{OH}^-] \) are both \( 1.0 \times 10^{-7} \) M.