To find the concentration of the unknown analyte based on its absorbance, we first need to establish a linear relationship between the concentrations and their corresponding absorbance values using the calibration data provided.
Step 1: Establish the linear relationship
From the calibration data provided, we observe a linear relationship where absorbance (A) is directly proportional to concentration (C) of the analyte.
We can express this relationship as: \[ A = mC + b \] where:
- \( A \) is the absorbance,
- \( C \) is the concentration,
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.
Step 2: Find the slope (m) and y-intercept (b)
The data points provided can be summarized as follows:
| Concentration (µM) | Absorbance | |---------------------|------------| | 1 | 0.12 | | 2 | 0.24 | | 3 | 0.36 | | 4 | 0.48 | | 5 | 0.60 |
Given that the relationship is linear, we can calculate the slope (m) and intercept (b) using two points from the data.
Using the first and last points (1 µM, 0.12) and (5 µM, 0.60):
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Calculate Slope (m): \[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} = \frac{0.60 - 0.12}{5 - 1} = \frac{0.48}{4} = 0.12 \]
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Calculate the y-intercept (b): We can use one of the points to solve for b. Using (1, 0.12): \[ 0.12 = 0.12(1) + b \implies b = 0.12 - 0.12 = 0 \] Thus, the equation of the line is: \[ A = 0.12C \]
Step 3: Use the equation to find concentration of the unknown
Given that the absorbance of the unknown is 45:
Using the linear equation: \[ 45 = 0.12C \]
To find \(C\): \[ C = \frac{45}{0.12} \approx 375 , \mu M \]
Step 4: Correlation Coefficient (r)
Since the relationship is linear, we can calculate the correlation coefficient (r) based on the data points.
To calculate r, you would ideally compute using the following formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \] where \(n\) is the number of points, \(x\) is the concentration, and \(y\) is the absorbance.
Calculating it step-by-step:
- Calculate sums:
- \(n = 5\)
- \(\sum x = 1 + 2 + 3 + 4 + 5 = 15\)
- \(\sum y = 0.12 + 0.24 + 0.36 + 0.48 + 0.60 = 1.8\)
- \(\sum xy = (10.12) + (20.24) + (30.36) + (40.48) + (5*0.60) = 0.12 + 0.48 + 1.08 + 1.92 + 3.00 = 6.6\)
- \(\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55\)
- \(\sum y^2 = (0.12^2) + (0.24^2) + (0.36^2) + (0.48^2) + (0.60^2) = 0.0144 + 0.0576 + 0.1296 + 0.2304 + 0.36 = 0.792\)
Substituting into the r formula: \[ r = \frac{5(6.6) - (15)(1.8)}{\sqrt{[5(55) - (15)^2][5(0.792) - (1.8)^2]}} \] \[ = \frac{33 - 27}{\sqrt{[275 - 225][3.96 - 3.24]}} = \frac{6}{\sqrt{50 \cdot 0.72}} = \frac{6}{\sqrt{36}} = \frac{6}{6} = 1 \]
Conclusion
- The concentration of the unknown analyte is approximately 375 µM.
- The correlation coefficient \( r \) is calculated to be 1, indicating a perfect linear relationship between concentration and absorbance in the data set. This suggests that the method is reliable and the calibration curve is valid over the range studied.