To determine the concentration of the unknown sample based on its absorbance, we can first analyze the calibration curve data provided. This data follows a linear relationship, so we can use linear regression to find the equation of the line that fits the standard data.
The concentration (x) is plotted against the absorbance (y) in the following way:
| Conc. (ppm) | Absorbance | |--------------|------------| | 1.5 | 0.15 | | 2.0 | 0.30 | | 2.5 | 0.45 | | 3.0 | 0.60 | | 3.5 | 0.75 | | 4.0 | 0.90 | | 4.5 | 1.05 | | 5.0 | 1.20 |
Step 1: Find the linear regression equation
We can represent this linear relationship with the equation of a line:
\[ y = mx + b \]
Where \( m \) is the slope and \( b \) is the y-intercept.
Step 2: Calculate the slope (m) and intercept (b)
The slope \( m \) can be calculated using the formula:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
The intercept \( b \) can be calculated as:
\[ b = \frac{\sum y - m(\sum x)}{N} \]
Where:
- \( N \) is the number of points,
- \( \sum xy \) is the sum of the product of x and y,
- \( \sum x \) is the sum of x values,
- \( \sum y \) is the sum of y values,
- \( \sum x^2 \) is the sum of squares of x values.
Step 3: Compute the necessary sums
Let's calculate the necessary sums step by step:
- \( \sum x = 1.5 + 2.0 + 2.5 + 3.0 + 3.5 + 4.0 + 4.5 + 5.0 = 22.0 \)
- \( \sum y = 0.15 + 0.30 + 0.45 + 0.60 + 0.75 + 0.90 + 1.05 + 1.20 = 5.30 \)
- \( \sum xy = (1.5)(0.15) + (2.0)(0.30) + (2.5)(0.45) + (3.0)(0.60) + (3.5)(0.75) + (4.0)(0.90) + (4.5)(1.05) + (5.0)(1.20) \)
Calculating \( \sum xy \):
\[ = 0.225 + 0.60 + 1.125 + 1.80 + 2.625 + 3.60 + 4.725 + 6.00 = 20.10 \]
- \( \sum x^2 = (1.5^2) + (2.0^2) + (2.5^2) + (3.0^2) + (3.5^2) + (4.0^2) + (4.5^2) + (5.0^2) \)
Calculating \( \sum x^2 \):
\[ = 2.25 + 4.00 + 6.25 + 9.00 + 12.25 + 16.00 + 20.25 + 25.00 = 95.00 \]
- Number of points \( N = 8 \)
Step 4: Plug into the formulas
Now, substituting into the slope formula:
\[ m = \frac{8(20.10) - (22.0)(5.30)}{8(95.00) - (22.0)^2} \] \[ = \frac{160.80 - 111.60}{760.00 - 484.00} = \frac{49.20}{276.00} \approx 0.178 \]
Now substituting \( m \) into the intercept formula:
\[ b = \frac{5.30 - (0.178)(22.0)}{8} \] \[ = \frac{5.30 - 3.916}{8} \approx \frac{1.384}{8} \approx 0.173 \]
Step 5: Equation of the calibration curve
Thus, the equation of the calibration curve is:
\[ y = 0.178x + 0.173 \]
Step 6: Determine the concentration of the unknown sample
Given the absorbance \( y = 0.65 \), we can solve for \( x \) (concentration):
\[ 0.65 = 0.178x + 0.173 \] \[ 0.65 - 0.173 = 0.178x \] \[ 0.477 = 0.178x \] \[ x = \frac{0.477}{0.178} \approx 2.68 \text{ ppm} \]
Step 7: Calculate the correlation coefficient (r)
The correlation coefficient \( r \) can be computed using the formula:
\[ r = \frac{N(\sum xy) - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}} \]
First, calculate \( \sum y^2 \):
\[ \sum y^2 = (0.15^2) + (0.30^2) + (0.45^2) + (0.60^2) + (0.75^2) + (0.90^2) + (1.05^2) + (1.20^2) = 0.0225 + 0.09 + 0.2025 + 0.36 + 0.5625 + 0.81 + 1.1025 + 1.44 = 4.675 \]
Now substitute into the correlation coefficient formula:
\[ r = \frac{8(20.10) - (22)(5.30)}{\sqrt{[8(95) - (22)^2][8(4.675) - (5.30)^2]}} \] \[ = \frac{160.80 - 116.60}{\sqrt{(760 - 484)(37.4 - 28.09)}} \] \[ = \frac{44.20}{\sqrt{276 \cdot 9.31}} \approx \frac{44.20}{\sqrt{2574.56}} \approx \frac{44.20}{50.74} \approx 0.87 \]
Conclusion
- The concentration of the unknown sample is approximately 2.68 ppm.
- The correlation coefficient \( r \) is approximately 0.87.
This indicates a strong positive linear relationship between concentration and absorbance.