Question

White wines tend to be more acidic than red wines. Find the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) in a Sauvignon Blanc with a \(\mathrm{pH}\) of 3.23 and a Cabernet Sauvignon with a pH of \(3.64 .\) How many times more acidic is the Sauvignon Blanc?

Short answer

The \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) in Sauvignon Blanc is \(10^{-3.23} = 5.75 \times 10^{-4} \,\mathrm{M}\) and in Cabernet Sauvignon is \(10^{-3.64} = 2.29 \times 10^{-4} \,\mathrm{M}\). Sauvignon Blanc is approximately \(10^{0.41}\) or 2.57 times more acidic than Cabernet Sauvignon.

Step-by-step solution

Understanding pH

The pH is a measure of the hydrogen ion concentration, where 'p' stands for 'potential' and 'H' stands for 'hydrogen.' The pH scale is logarithmic, which means each whole pH value below 7 (the neutral pH) is ten times more acidic than the next higher value. The formula to find the hydrogen ion concentration \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) from the pH is given by \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 10^{-\mathrm{pH}}\).

Calculating \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) for Sauvignon Blanc

Insert the pH value of the Sauvignon Blanc into the formula to calculate its hydrogen ion concentration: \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 10^{-3.23}\).

Calculating \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) for Cabernet Sauvignon

Use the same process for the Cabernet Sauvignon by inserting its pH value into the formula: \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 10^{-3.64}\).

Compare the Acidities

To find how many times more acidic the Sauvignon Blanc is compared to the Cabernet Sauvignon, divide the hydrogen ion concentration of the Sauvignon Blanc by that of the Cabernet Sauvignon: \(\frac{10^{-3.23}}{10^{-3.64}}\).

Simplify the Expression

Using the properties of exponents, the expression simplifies to \(10^{3.64 - 3.23}\), which equals \(10^{0.41}\). The result will give the factor by which the acidity of Sauvignon Blanc is greater than that of Cabernet Sauvignon.

Key concepts

pH Scale

The pH scale is an essential tool in chemistry that enables us to measure how acidic or basic a solution is. It ranges from 0 to 14, with lower numbers indicating high acidity, higher numbers indicating high alkalinity, and 7 being neutral, like pure water.

To understand this better, picture the pH scale as a spectrum where each whole number change represents a tenfold difference in acidity or alkalinity. For example, a solution with a pH of 4 is ten times more acidic than one with a pH of 5, and a hundred times more acidic than a solution with a pH of 6. This logarithmic scale is what makes the pH a very sensitive measure of the hydrogen ion concentration in solutions.

Hydrogen Ion Concentration

In chemistry, the acidity of a solution is determined by the concentration of hydrogen ions, denoted as \( \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \). These ions are formed when a substance that can donate hydrogen ions is dissolved in water.

In the context of wine, as with the Sauvignon Blanc and Cabernet Sauvignon example, the concentration of these ions dictates the tartness or sharp taste you experience. The lower the pH, the higher the concentration of hydrogen ions, and consequently, the more acidic the wine. Measuring this concentration provides a precise way to compare the acidity of different beverages or solutions.

Logarithmic Scale

The pH scale is based on a logarithmic scale, which means it's a non-linear scale used for a wide range of values. A logarithmic scale is different from a linear scale, where equal distances represent equal increments. Instead, each increment on the pH scale increases by a factor of ten because of its base-10 logarithmic nature.

For any given pH value, the hydrogen ion concentration of the solution is determined by the negative exponent of 10. So, if a wine has a pH of 3, the actual concentration of hydrogen ions is \( 10^{-3} \) moles per liter. This logarithmic nature makes it easy to work with the very large or small numbers that can arise when measuring hydrogen ion concentrations.

Properties of Exponents

Exponents are a convenient way to express repeated multiplication, and they play a crucial role in calculating and comparing pH values. When comparing the acidity of two solutions, we often end up dividing one exponential expression by another.

In our example, the exponent laws allow us to simplify the comparison of \( \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \) between Sauvignon Blanc and Cabernet Sauvignon by subtracting the exponents. This simplification is possible because of the property that states \( a^{m} \div a^{n} = a^{m-n} \), where \( a \) is a non-zero base and \( m \) and \( n \) are exponents. Understanding this property is vital because it enables us to quickly see that one solution is \( 10^{0.41} \) times more acidic than another without tedious calculation.