Question

Calculate the concentration of \(\mathrm{HBr}\) in a solution at \(25^{\circ} \mathrm{C}\) that has a pH of (a) 0.12 , (b) \(2.46,\) and \((\mathrm{c}) 6.27\).

Short answer

(a) 0.757 M, (b) 3.47 x 10^-3 M, (c) 5.37 x 10^-7 M

Step-by-step solution

Understanding pH and Its Relationship to \\([ ext{H}^+]\\)

The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration: \[ \text{pH} = -\log [\text{H}^+] \] To find the concentration of \(\text{H}^+\) in the solution, we must rearrange the formula to solve for \([\text{H}^+]\):\[ [\text{H}^+] = 10^{-\text{pH}} \]

Calculating [H⁺] for each pH value

Convert the given pH values into \([\text{H}^+]\) concentrations using the formula derived in Step 1: - For pH = 0.12: \[ [\text{H}^+] = 10^{-0.12} \approx 0.757 \, \text{M} \]- For pH = 2.46: \[ [\text{H}^+] = 10^{-2.46} \approx 3.47 \times 10^{-3} \, \text{M} \] - For pH = 6.27: \[ [\text{H}^+] = 10^{-6.27} \approx 5.37 \times 10^{-7} \, \text{M} \]

Relating [H⁺] to [HBr]

Since \(\mathrm{HBr}\) is a strong acid, it dissociates completely in water. This means the concentration of \(\text{H}^+\) ions is equal to the concentration of \(\mathrm{HBr}\) in the solution. Thus, \([\mathrm{HBr}] = [\text{H}^+]\).

Conclusion

For each pH value, the concentration of \(\mathrm{HBr}\) is equal to the calculated \([\text{H}^+]\):- For pH = 0.12, \([\mathrm{HBr}] \approx 0.757 \, \text{M}\)- For pH = 2.46, \([\mathrm{HBr}] \approx 3.47 \times 10^{-3} \, \text{M}\)- For pH = 6.27, \([\mathrm{HBr}] \approx 5.37 \times 10^{-7} \, \text{M}\)

Key concepts

Hydrogen Ion Concentration

Hydrogen ion concentration, represented as \([\text{H}^+]\), is key to understanding the acidity of a solution. It measures how many hydrogen ions are present in a given amount of solution. The concentration of hydrogen ions affects the pH level, which we calculate using the formula: \[ \text{pH} = -\log [\text{H}^+] \] Being comfortable with this formula is vital for calculating pH or converting pH back to \([\text{H}^+]\). Simply rearrange the equation to find the concentration: \[ [\text{H}^+] = 10^{-\text{pH}} \] This formula lets you determine how many hydrogen ions are present in various solutions.

Strong Acid Dissociation

Strong acids, like HBr, dissociate completely in water. This means that when HBr is added to water, it breaks down entirely into hydrogen ions \([\text{H}^+]\) and bromide ions \([\text{Br}^-]\). In other words, the concentration of \([\text{H}^+]\) directly equals the original concentration of HBr. This relationship simplifies pH calculations:

• For every 1 mole of HBr, there will be 1 mole of \([\text{H}^+]\) in solution.
• Thus, the hydrogen ion concentration determined from pH directly tells us the concentration of HBr.

Understanding this helps make solving pH-related problems much easier.

Logarithmic Functions

Logarithmic functions play a crucial role in chemistry, especially in pH calculations. The pH scale is logarithmic, meaning each whole number step represents a tenfold change. For instance, a pH of 3 is ten times more acidic than a pH of 4. When working with logarithms:

• \(\log(10^x) = x\): The logarithm of a power of 10 returns the exponent.
• Performs inverse operations to extract \([\text{H}^+]\) using \(10^{-\text{pH}}\).

Grasping this concept is essential for translating between pH and \([\text{H}^+]\) concisely.

Solution Concentration

The concentration of a solution tells us the amount of solute present in a given volume. In this context, we're interested in how much HBr is in our solution, which directly equates to the hydrogen ion concentration due to complete dissociation. Several factors affect concentration:

• Temperature can influence solubility and reaction rates, though here it's consistent at \(25^{\circ}C\).
• The relationship defined by strong acid dissociation ensures \([\text{HBr}] = [\text{H}^+]\).

Knowing the principles of solution concentration assists in comprehending how dilute or concentrated a solution may be based on its pH.