Question

Consider two solutions, solution A and solution B. \(\left[\mathrm{H}^{+}\right]\) in solution \(A\) is 250 times greater than that in solution \(B\). What is the difference in the pH values of the two solutions?

Short answer

The difference in the pH values of the two solutions is approximately 2.4.

Step-by-step solution

Define the relationship

Given that the hydrogen ion concentration in solution A is 250 times greater than in solution B, we can write the relationship as: \[\left[\mathrm{H}_{A}^{+}\right] = 250 \left[\mathrm{H}_{B}^{+}\right]\]

Calculate the pH values for both solutions

Firstly, let's use the formula for pH, which is pH = -log(\([\mathrm{H}^{+}\])) for both solutions. For Solution A: pH(A) = -log(\([\mathrm{H}_{A}^{+}\])) For Solution B: pH(B) = -log(\([\mathrm{H}_{B}^{+}\]))

Calculate the difference in pH values

The difference in the pH values of the two solutions is given by the absolute difference of their pH: \[\Delta \textrm{pH} = | \textrm{pH(A)} - \textrm{pH(B)} |\] To find this difference, we can substitute our expressions for pH(A) and pH(B) from step 2: \[\Delta \textrm{pH} = | -\textrm{log}(\left[\mathrm{H}_{A}^{+}\right]) + \textrm{log}(\left[\mathrm{H}_{B}^{+}\right]) |\]

Use the logarithmic properties to simplify the expression

We can use the logarithmic properties to simplify our expression for the difference in pH. Specifically, we will use the property: log(a) - log(b) = log(a/b) Applying this property, we have: \[\Delta \textrm{pH} = | \textrm{log}\left(\frac{\left[\mathrm{H}_{B}^{+}\right]}{\left[\mathrm{H}_{A}^{+}\right]}\right) |\] Now, using the relationship we found in step 1, we can replace \([\mathrm{H}_{A}^{+}\]) with \(250 \left[\mathrm{H}_{B}^{+}\right]\) to get: \[\Delta \textrm{pH} = | \textrm{log}\left(\frac{\left[\mathrm{H}_{B}^{+}\right]}{250\left[\mathrm{H}_{B}^{+}\right]}\right) |\]

Simplify and compute the result

Now, we can cancel out the \([\mathrm{H}_{B}^{+}\]) terms in the numerator and denominator to get: \[\Delta \textrm{pH} = | \textrm{log}\left(\frac{1}{250}\right) |\] Since the logarithmic function is strictly decreasing, taking the absolute value of the logarithm of a number less than 1 will simply change the sign of the result. Therefore, we have: \[\Delta \textrm{pH} = \textrm{log}\left(250\right) \] To get the final result, we simply compute the logarithm: \[\Delta \textrm{pH} \approx 2.4\] So, the difference in the pH values of the two solutions is approximately 2.4.

Key concepts

Hydrogen Ion Concentration

Understanding the hydrogen ion concentration is crucial in chemistry, especially when discussing the acidity or basicity of a solution. The hydrogen ion concentration, denoted as \( [H^+] \), is a measure of the number of hydrogen ions present in a solution. It's a key factor in determining the pH of the solution.

When it comes to calculating the pH difference between two solutions, it's important to know that the hydrogen ion concentration of one solution can be a multiple of that of another. In the exercise, solution A has a hydrogen ion concentration that is 250 times greater than that of solution B. Intuitively, this means solution A is significantly more acidic than solution B.

The higher the concentration of hydrogen ions, the lower the pH value, which leads to a more acidic solution. Conversely, a lower hydrogen ion concentration corresponds to a higher pH value, indicating a more basic solution. Recognizing this inverse relationship is essential when interpreting pH differences between solutions and navigating the pH scale effectively.

Logarithmic Properties

Logarithms are a mathematical tool used to understand and manipulate powers of numbers, and they play an integral role in pH calculations. A logarithm, specifically the common logarithm used for pH calculations, tells us what power we need to raise 10 to in order to get a specific number. The formula for pH is a practical application of logarithms: \( pH = -\log([H^+]) \).

One of the logarithmic properties used in the provided exercise is the subtraction rule: \(\log(a) - \log(b) = \log(a/b)\). This rule helps simplify the calculations when trying to find the difference between two pH values. It's a powerful tool because it allows us to convert the ratio of two hydrogen ion concentrations into a simple logarithmic expression, which can be easily evaluated to find the pH difference.

This logarithmic property effectively bridges the gap between the raw concentration data and the final pH calculation. It's through leveraging properties like this that we are able to perform complex calculations in a clear and concise manner.

The pH Scale

The pH scale is a numerical scale ranging from 0 to 14, which is used to specify the acidity or basicity of an aqueous solution. At room temperature, a pH value of 7 is considered neutral, with values below 7 indicating acidity and values above 7 indicating basicity. The pH scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic than the next higher value.

In the context of our exercise, understanding this scale is pivotal. Knowing that the pH of solution A is significantly lower than the pH of solution B allows us to predict that solution A must have a much higher concentration of hydrogen ions. The scale is not linear; a small pH difference can represent a huge change in hydrogen ion concentration, as evidenced by the fact that a 2.4 difference in pH between the solutions corresponds to one being 250 times more acidic than the other.

Concisely, the pH scale provides not only a measure of relative acidity or basicity but also an insight into the hydrogen ion concentration in a very practical way. It's an essential concept for students to master, to fully understand the nature of solutions they work with in chemistry.