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Chapter 4: Problem 29

These exercises deal with logarithmic scales. The pH reading of a sample of each substance is given. Calculate the hydrogen ion concentration of the substance. (a) Vinegar: \(\mathrm{pH}=3.0\) (b) Milk: \(\mathrm{pH}=6.5\)

Short Answer

Expert verified
Vinegar: \(1 \times 10^{-3}\) M; Milk: \(3.16 \times 10^{-7}\) M.

Step by step solution

01

Understanding pH and Hydrogen Ion Concentration

The pH scale is used to specify the acidity or basicity of an aqueous solution. It is a logarithmic scale, and the pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration: \[\text{pH} = -\log_{10}[H^+]\]where \([H^+]\) is the concentration of hydrogen ions in moles per liter.
02

Rearranging the pH Formula

To find the hydrogen ion concentration from the pH value, rearrange the formula: \[[H^+] = 10^{-\text{pH}}\]This allows us to calculate the concentration by taking the power of 10 to the negative pH value.
03

Calculating Hydrogen Ion Concentration for Vinegar

Given the pH of vinegar is 3.0, substitute this value into the rearranged formula:\[[H^+] = 10^{-3.0}\]Thus, the hydrogen ion concentration for vinegar is \[[H^+] = 1 imes 10^{-3}\text{ M (Moles per Liter)}\].
04

Calculating Hydrogen Ion Concentration for Milk

Given the pH of milk is 6.5, substitute this value into the rearranged formula:\[[H^+] = 10^{-6.5}\]Thus, the hydrogen ion concentration for milk is \[[H^+] = 3.16 imes 10^{-7}\text{ M (Moles per Liter)}\].

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Key Concepts

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

pH scale
The pH scale is a crucial tool used in chemistry to determine how acidic or basic a solution is. It operates on a range from 0 to 14, allowing scientists to classify substances from highly acidic to highly basic.
The scale is logarithmic, meaning each whole-number change on the pH scale represents a tenfold change in hydrogen ion concentration.
  • A pH less than 7 indicates an acidic solution, like vinegar or lemon juice.
  • A pH of 7 is considered neutral, as seen in pure water.
  • A pH greater than 7 signifies a basic or alkaline solution, such as baking soda.
The pH scale's logarithmic nature makes it particularly useful. This is because it can accommodate the wide range of hydrogen ion concentrations found in different substances. In essence, the scale provides a simplified way of expressing these concentrations, making it easier to understand and compare different solutions.
Hydrogen Ion Concentration
Hydrogen ion concentration, often denoted as \([H^+]\), is a measure of the amount of hydrogen ions present in a solution. It is expressed as the number of moles of hydrogen ions per liter of solution (M).A higher concentration of hydrogen ions means a lower pH and indicates a more acidic solution. Conversely, a lower concentration of hydrogen ions implies a higher pH and a more basic solution.
Understanding hydrogen ion concentration helps in various applications, such as:
  • Determining the acidity level of food and beverages, like vinegar or milk.
  • Monitoring the pH balance in pools, aquariums, and even the human body for health reasons.
Effectively, the hydrogen ion concentration directly impacts the chemical behavior and biological functionality of substances.
Logarithmic Functions
Logarithmic functions play a pivotal role in the mathematical representation of the pH scale. A logarithmic function, in general, is the inverse of an exponential function. When related to pH, the expression \(\text{pH} = -\log_{10}[H^+]\) shows how the pH depends on the hydrogen ion concentration.This logarithmic expression conveniently transforms the vast range of hydrogen ion concentrations into a more manageable numerical range.
The characteristics include:
  • Handling large variations in data, like hydrogen ion concentrations, by significantly compressing the range of values.
  • Providing a simple method to calculate pH by rearranging the formula: \([H^+] = 10^{-\text{pH}}\). This helps in calculations involving pH measurements.
Thus, logarithmic functions allow for complex and wide-ranging data to be easily understood and used in practical applications, making them invaluable in scientific calculations.

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Most popular questions from this chapter

Wealth Distribution Vilfredo Pareto \((1848-1923)\) observed that most of the wealth of a country is owned by a few members of the population. Pareto's Principle is $$\log P=\log c-k \log W$$ where \(W\) is the wealth level (how much money a person has) and \(P\) is the number of people in the population having that much money. (a) Solve the equation for \(P\) (b) Assume \(k=2.1, c=8000,\) and \(W\) is measured in millions of dollars. Use part (a) to find the number of people who have \(\$ 2\) million or more. How many people have \(\$ 10\) million or more?

Shifting, Shrinking, and Stretching Graphs of Functions Let \(f(x)=x^{2} .\) Show that \(f(2 x)=4 f(x),\) and explain how this shows that shrinking the graph of \(f\) horizontally has the same effect as stretching it vertically. Then use the identities \(e^{2+x}=e^{2} e^{x}\) and \(\ln (2 x)=\ln 2+\ln x\) to show that for \(g(x)=e^{x},\) a horizontal shift is the same as a vertical stretch and for \(h(x)=\ln x,\) a horizontal shrinking is the same as a vertical shift.

These exercises deal with logarithmic scales. Inverse Square Law for Sound A law of physics states that the intensity of sound is inversely proportional to the square of the distance \(d\) from the source: \(I=k / d^{2}\). (a) Use this model and the equation $$B=10 \log \frac{I}{I_{0}}$$ (described in this section) to show that the decibel levels \(B_{1}\) and \(B_{2}\) at distances \(d_{1}\) and \(d_{2}\) from a sound source are related by the equation $$B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}}$$ (b) The intensity level at a rock concert is 120 dB at a distance of 2 m from the speakers. Find the intensity level at a distance of 10 m.

The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model $$ n(t)=\frac{5600}{0.5+27.5 e^{-0.044 t}} $$ where \(t\) is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function \(n(t)\) (c) What size does the population approach as time goes on?

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$\log _{4} 125$$
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