Question

Calculate the \(\mathrm{pH}\) of a solution whose hydrogen ion concentration is \(2.7 \times 10^{-6} \mathrm{M}\).

Short answer

The pH of a solution with a hydrogen ion concentration of \(2.7 \times 10^{-6} M\) can be calculated using the formula \(pH = -\log[H^+]\), where \(H^+\) is the hydrogen ion concentration. Substituting the given value, we get \(pH = -\log(2.7 \times 10^{-6}) \approx -(-5.57)\), which gives us a pH value of approximately 5.57.

Step-by-step solution

Remember the pH formula

The pH is calculated using the following formula: \[ pH = -\log[H^+] \] where \( pH \) is the pH of the solution, \( H^+ \) is the hydrogen ion concentration, and \( \log \) represents the logarithm base 10.

Substitute the hydrogen ion concentration

Substitute the given hydrogen ion concentration, \(2.7 \times 10^{-6} M\), into the pH formula: \[ pH = -\log(2.7 \times 10^{-6}) \]

Calculate the pH

Use a calculator or logarithm table to find the value of the logarithm and calculate the pH: \[ pH = -\log(2.7 \times 10^{-6}) \approx -(-5.57) \]

Find the final pH value

Based on the calculation, the pH of the solution is approximately: \[ pH \approx 5.57 \] Therefore, the pH of the solution is approximately 5.57.

Key concepts

pH formula

Understanding the pH formula is fundamental for anyone looking to grasp the acidity or basicity of a solution. The formula itself is quite elegant and simple:
\[ pH = -\log[H^+] \]
This equation connects the pH of a solution directly to the hydrogen ion concentration, \( [H^+] \), using the logarithmic function. The negative sign ensures that a higher concentration of ions, which normally indicates a more acidic solution, results in a lower pH value. This is consistent with the scale where pH levels range from 0 (most acidic) to 14 (most basic). Normal water at room temperature has a pH of approximately 7, which is considered neutral.

Hydrogen ion concentration

Hydrogen ion concentration is the measure of the amount of hydrogen ions present in a solution. It’s typically expressed in moles per liter (M), which can also be thought of as the 'strength' of the acid or base in question. A higher hydrogen ion concentration means more acidic the solution is. It's this very concentration that plays a key part in the formula to calculate pH. For instance, a hydrogen ion concentration of \(2.7 \times 10^{-6} M\) implies that there are 2.7 moles of hydrogen ions in every liter of solution. This is a relatively low concentration, which would correspond to a pH that is on the acidic side, yet closer to neutral compared to stronger acids.

Logarithm

A logarithm is a mathematical operation that tells us the power to which a number must be raised to obtain another number. Specifically, when we talk about pH, we're referring to the base 10 logarithm. It’s useful to know that the logarithm of a number that's written in scientific notation, such as \(2.7 \times 10^{-6}\), simplifies the process of finding pH. We take the negative logarithm of the hydrogen ion concentration to calculate the pH, which effectively inverts the exponential relationship between the two values. In practice, when calculating the pH for \(2.7 \times 10^{-6} M\), the negative sign in the formula cancels out the negative from the exponent in the scientific notation, thus leaving us with a positive pH value.

Properties of acids and bases

Acids are substances that increase the concentration of hydrogen ions \( (H^+) \) when dissolved in water, while bases decrease hydrogen ion concentration in the same context. Understanding the properties of acids and bases is essential when discussing pH. Acids have sour taste and can react strongly with metals, often releasing hydrogen gas, while bases are known for their bitter taste and slippery feel. The pH scale, ranging from 0 to 14, is a direct way to denote their acidity or basicity. Acids have a pH less than 7, bases have a pH greater than 7, and a neutral substance like pure water has a pH of exactly 7. It's these chemical properties that play a significant role in a wide range of scientific and industrial processes, from biology to food science to manufacturing.