Question

Applications. \(p H:\) The \(\mathrm{pH}\) value of a solution having a concentration \(C\) of hydrogen ions is given by the following equation: $$\begin{array}{|llll} \text { pH } & \text { pH }=-\log _{10} C & 1048 \end{array}$$ The units of \(C\) are moles of hydrogen ions per liter of solution. A mole is a unit used in chemistry for expressing large quantities of very small entities. A mole of hydrogen ions, for example, has a mass of one gram. Find the concentration of hydrogen ions in a solution having a pH of 4.65.

Short answer

The concentration of hydrogen ions in a solution with a pH of 4.65 is \(C = 10^{-4.65} \approx 2.24 \times 10^{-5}\) moles per liter.

Step-by-step solution

Understand the pH Equation

The pH of a solution is calculated using the formula \(\text{pH} = -\log_{10} C\), where \(C\) is the concentration of hydrogen ions in moles per liter. To find the concentration when given the pH, we need to solve for \(C\).

Set up the Equation Using the Given pH

Given that the pH of the solution is 4.65, we substitute this value into the pH equation: \(-\log_{10} C = 4.65\).

Solve for Concentration

To find the concentration \(C\), we need to rewrite the equation in exponential form. The inverse of the logarithm is the exponential function, so we have \(10^{-4.65} = C\).

Calculate the Concentration

Compute the concentration by evaluating the exponential expression: \(-\log_{10} C = 4.65\) implies \(C = 10^{-4.65}\). To find the value of \(10^{-4.65}\), use a calculator.

Express the Final Answer

After calculating the exponent, we find the concentration of hydrogen ions in the solution, which will be in moles per liter (M).

Key concepts

Hydrogen Ion Concentration

Understanding the concentration of hydrogen ions, or protons, in a solution is a critical aspect of acid-base chemistry. Essentially, it's a measure of how acidic or alkaline a solution is. In scientific terms, hydrogen ion concentration is expressed in moles per liter of solution. This unit of concentration can seem abstract at first, but it's analogous to other measurement concepts, like measuring sugar in packets per cup of coffee.

For a practical illustration, consider a lemon's juice, which is acidic and therefore has a higher hydrogen ion concentration compared to a base like baking soda solution. It's the abundance of these microscopic charged particles that gives substances their characteristic pH levels.

Logarithmic Functions

Logarithmic functions play a vital role in understanding various scientific and mathematical concepts, including pH calculation in chemistry. A logarithm, in simple terms, is the opposite of an exponent. Think of it like a detective working to find out 'whodunit' when given a clue in the form of an exponent. If we consider a number like 100, which can be written as 10^2 (10 to the power of 2), the logarithm asks '10 raised to what power gives you 100?' And the answer would be 2.

In pH calculations, we use a base 10 logarithm because it aligns with our decimal system, making it easier to comprehend changes in hydrogen ion concentrations. The negative sign in the pH formula indicates that as hydrogen ion concentration increases, pH values decrease, meaning the solution becomes more acidic.

Exponential Expressions

Exponential expressions, which are often seen in the context of growth or decay in nature, such as population sizes or radioactive substance decay, are also fundamental in pH calculations. When we talk about the exponential expression 10^-4.65, it represents a very small number, because any positive number raised to a negative power is a fraction.

Interpreting these numbers can become intuitive with a bit of practice. For instance, a pH of 7, which corresponds to a neutral solution, would calculate to an exponential expression of 10^-7, perfectly centering it on the pH scale that ranges from 0 to 14. As the exponent becomes smaller (more negative), the pH value lowers and the acidity increases.

Acid-Base Chemistry

The very heart of acid-base chemistry lies in the exchange and concentration of hydrogen ions. When an acid dissolves in water, it donates hydrogen ions, increasing the hydrogen ion concentration and thus lowering the pH. Conversely, a base accepts hydrogen ions, lowering their concentration in the solution and consequently raising the pH.

This process is fundamental to many natural phenomena and technological applications. For example, human blood maintains a very narrow pH range around 7.4, which is crucial for proper physiological functions. Similarly, understanding and manipulating pH levels are essential in industries ranging from agriculture, where soil pH affects plant growth, to pharmaceuticals, where precise pH control can affect drug effectiveness and safety.