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A sample of lemon juice has a hydronium-ion concentration equal to \(2.5 \times 10^{-2} M .\) What is the \(\mathrm{pH}\) of this sample?

Short Answer

Expert verified
The pH of the lemon juice sample is approximately 1.60.

Step by step solution

01

Understand the Formula for pH

The \(\text{pH}\) of a solution is calculated using the formula: \(\text{pH} = -\log[H_3O^+]\), where \([H_3O^+]\) is the concentration of hydronium ions in the solution.
02

Substitute the Value into the Formula

We are provided with the hydronium-ion concentration \( [H_3O^+] = 2.5 \times 10^{-2} M \). Substitute this value into the \(-\log\) formula: \(-\log(2.5 \times 10^{-2})\).
03

Calculate the pH Using Logarithms

Calculate \(-\log(2.5 \times 10^{-2})\). This calculation typically involves using a calculator. The \(\log(2.5)\) part equals approximately 0.3979, and \(\log(10^{-2}) = -2\). Add these values together to find the \(-\log(2.5 \times 10^{-2})\).
04

Simplify the Expression

Combine the logarithmic values: \(-\log(2.5) = -0.3979\) and \(-\log(10^{-2}) = 2\). Therefore, \(-\log(2.5 \times 10^{-2}) = 2 - 0.3979\).
05

Final Calculation

Perform the final calculation: \(2 - 0.3979 = 1.6021\). Therefore, the pH of the lemon juice sample is approximately \(\text{pH} = 1.60\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acid-Base Chemistry
Acid-base chemistry is an essential part of chemistry that helps explain the behavior of substances in various solutions. In simple terms, acids and bases are substances that can donate or accept protons, respectively. Understanding this concept is crucial when studying pH, a scale that measures the acidity or basicity of a solution.
A solution with a pH less than 7 is considered acidic, while a pH greater than 7 is considered basic or alkaline. For example, lemon juice is acidic because it has a high concentration of hydronium ions. A neutral solution, like pure water, typically has a pH of exactly 7.
  • An acid increases the hydronium ions ([H鈧僌鈦篯) in a solution, leading to a lower pH.
  • A base decreases the hydronium ions (or increases hydroxide ions, [OH鈦籡), resulting in a higher pH.
In essence, by understanding the relationship between hydronium ion concentration and pH, one can easily determine whether a solution is acidic or basic.
Hydronium Ion Concentration
The concentration of hydronium ions in a solution is a key indicator of its acidity. Hydronium ions are formed when an acid dissolves in water and donates a proton to a water molecule ( H鈧侽 to form H鈧僌鈦). This process is expressed by the equation: H鈦 + H鈧侽 鈫 H鈧僌鈦.
The concentration of hydronium ions, [H鈧僌鈦篯 , can be directly converted into pH, which tells us how acidic or basic a solution is. For example, the lemon juice mentioned in the exercise has [H鈧僌鈦篯 = 2.5 脳 10鈦宦 M. High concentrations of H鈧僌鈦 ions correspond to more acidic solutions and thus lower pH values.
  • To find pH: Use the formula pH = - log [H鈧僌鈦篯.
  • High [H鈧僌鈦篯 means low pH (acidic), while low [H鈧僌鈦篯 means high pH (basic).
By understanding hydronium ion concentration, we can quantify the extent of a solution鈥檚 acidity.
Logarithmic Functions
Logarithmic functions are mathematical operations that are widely used in chemistry, particularly in pH calculations. A logarithm answers the question: "To what power must a given base be raised, to produce a certain number?" In pH calculations, we use base 10 logarithms, which simplify the relationship between hydronium ion concentration and pH.
When calculating the pH, you will often encounter the formula: a = - log b , where 'b' is the concentration of hydronium ions. The log function helps scale the vast range of possible ion concentrations into a more manageable pH range, typically between 0 and 14. This transformation often requires a calculator, as it deals with exponents and fractions.
  • Use log to find pH: Calculate -log( [H鈧僌鈦篯 ) to determine how acidic a solution is.
  • The log scale is nonlinear: each whole number change in pH represents a tenfold change in hydronium ion concentration.
By understanding logarithms, students can better grasp the concept of pH and handle the large disparities in ion concentrations.

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