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Chapter 10: Problem 40

The following data is recorded for the phosphorescent intensity of several standard solutions of benzo[a] pyrene. $$ \begin{array}{cc} \text { [benzo[a]pyrene] }(\mathrm{M}) & \text { emission intensity } \\ \hline 0 & 0.00 \\ 1.00 \times 10^{-5} & 0.98 \\ 3.00 \times 10^{-5} & 3.22 \\ 6.00 \times 10^{-5} & 6.25 \\ 1.00 \times 10^{-4} & 10.21 \end{array} $$ What is the concentration of benzo[a] pyrene in a sample that yields a phosphorescent emission intensity of \(4.97 ?\)

Short Answer

Expert verified
The concentration of benzo[a]pyrene is approximately \(4.87 \times 10^{-5} \text{ M}\).

Step by step solution

01

Understand the Relationship

The data reflects a relationship between the concentration of benzo[a]pyrene and the phosphorescent emission intensity. We can infer from the data that this relationship is linear because the increase in intensity appears to be proportional to the increase in concentration.
02

Construct the Equation of the Line

Using the standard form of the linear equation: \[ y = mx + c \]we need to determine the slope \(m\) and the y-intercept \(c\). Using the points \((0, 0)\) and \((1.00 \times 10^{-4}, 10.21)\) to determine the slope:\[ m = \frac{10.21 - 0}{1.00 \times 10^{-4} - 0} = 102100 \]Thus, the equation is:\[ y = 102100x \]
03

Solve for the Unknown Concentration

Given the phosphorescent emission intensity is 4.97, substitute \(y = 4.97\) in the equation:\[ 4.97 = 102100x \]Solve for \(x\):\[ x = \frac{4.97}{102100} \approx 4.87 \times 10^{-5} \]
04

Verify the Solution

Verify by substituting \(x = 4.87 \times 10^{-5}\) back into the intensity equation to check if it approximates 4.97:\[ y = 102100 \times (4.87 \times 10^{-5}) \approx 4.97 \]Since the calculated intensity closely matches the given intensity, the solution is verified.

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Key Concepts

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

Benzo[a]pyrene
Benzo[a]pyrene is a type of polycyclic aromatic hydrocarbon (PAH) that is known for its complex structure. It consists of multiple benzene rings fused together, forming a flat and sturdy arrangement. This compound is often studied due to its presence in the environment as a pollutant, produced during the incomplete combustion of organic materials.
In analytical chemistry, we aim to detect and measure benzo[a]pyrene because it's considered a carcinogen. This detection is crucial for environmental monitoring and health assessments.
Given its environmental impact, understanding how to detect its concentration using scientific techniques is a valuable skill.
Phosphorescent Emission
Phosphorescent emission is a type of luminescence that lasts longer than fluorescence. When a molecule like benzo[a]pyrene absorbs energy, its electrons get excited to a higher energy state. When they return to their original state, they emit light.
What makes phosphorescence unique is the "slow" return of these electrons. Due to this delay, phosphorescent materials continue to glow even after the exciting source is removed.
Benzo[a]pyrene's phosphorescent properties are used in analytical methods to quantify its concentration in various samples. It's an effective way to track less visible materials by observing their glow.
Concentration Calculation
Calculating the concentration of a substance like benzo[a]pyrene involves understanding its relationship with a measurable parameter 鈥� in this case, phosphorescent emission intensity.
To determine concentration, we can apply the equation of a line, calculated from given data points. This involves knowing the slope (\[m\] of the line) and using it to find the unknown concentration (\[x\]).
The equation, derived from linear data, helps us find the exact amount of benzo[a]pyrene in our sample. This method is crucial in environmental chemistry, where precise measurements are often necessary for safety and compliance.
Linear Relationship in Chemistry
A linear relationship is a key concept in many chemistry applications as it simplifies the prediction of one variable based on another. When dealing with benzo[a]pyrene, the phosphorescent intensity is directly proportional to its concentration; hence, the relationship is linear.
Mathematically, a linear relationship is expressed with the equation \[y = mx + c\]. Here, \[y\] represents the dependent variable (emission intensity), \[x\] the independent variable (concentration), \[m\] the slope, and \[c\] the y-intercept.
The linear trend suggests that as benzo[a]pyrene's concentration increases, so does the phosphorescent emission intensity, perfectly aligning with the data. Such predictable patterns are fundamental in analytical chemistry, facilitating quantification using straightforward calculations.

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