I get how to use the method of least squares to determine slope, intercept, uncertainties in slope and intercept of your graph and but am confused how to determine the error of mole percent stated in (ii)
The following are peak areas for chromatograms of isooctane standards
Peak area Mole percent isotane
0.352 1.09
0.803 1.78
1.08 2.60
1.38 3.03
1.75 4.01
(i)Use the method of least squares to find the slope, intercept, uncertainties in slope and intercept of your graph and give the equation of the line.
(ii) A peak area of 2.65 was obtained for an unknown sample of hydrocarbon mixture. Calculate the mole percent of isooctane in the mixture, error of the mole percent and the 95% confidence interval of the mole percent if the peak area was
(a) the result of a single measurement and
(b) the mean of four measurements
(c) comment on the difference in answers to (a) and (b)
Hope anyone can help me with this homework
To determine the error of the mole percent stated in part (ii), you'll need to use the equation of the line obtained from part (i). Let's recap the information we have:
Peak areas for isooctane standards:
- Peak area: 0.352, Mole percent isooctane: 1.09
- Peak area: 0.803, Mole percent isooctane: 1.78
- Peak area: 1.08, Mole percent isooctane: 2.60
- Peak area: 1.38, Mole percent isooctane: 3.03
- Peak area: 1.75, Mole percent isooctane: 4.01
(i) To find the slope, intercept, uncertainties in slope and intercept of the graph, you should perform a linear regression analysis using the method of least squares. This will allow you to find the equation of the line that best fits the data points. Here are the general steps to follow:
1. Calculate the average peak area and average mole percent isooctane for the given data points.
Average peak area = (0.352 + 0.803 + 1.08 + 1.38 + 1.75) / 5 = 1.0736
Average mole percent isooctane = (1.09 + 1.78 + 2.60 + 3.03 + 4.01) / 5 = 2.702
2. Calculate the deviations of each data point from the respective average value.
Deviation of peak area = Peak area - Average peak area
Deviation of mole percent isooctane = Mole percent isooctane - Average mole percent isooctane
Deviations:
(0.352 - 1.0736), (1.09 - 2.702) = -0.7216, -1.612
(0.803 - 1.0736), (1.78 - 2.702) = -0.2706, -0.922
(1.08 - 1.0736), (2.60 - 2.702) = 0.0064, -0.102
(1.38 - 1.0736), (3.03 - 2.702) = 0.3064, 0.328
(1.75 - 1.0736), (4.01 - 2.702) = 0.6764, 1.308
3. Calculate the squares of the deviations.
Squares of deviations:
(-0.7216)^2, (-1.612)^2 = 0.5218, 2.5952
(-0.2706)^2, (-0.922)^2 = 0.0732, 0.8493
(0.0064)^2, (-0.102)^2 = 0.000041, 0.0105
(0.3064)^2, (0.328)^2 = 0.0939, 0.1074
(0.6764)^2, (1.308)^2 = 0.457, 1.712
4. Calculate the sum of the squares of the deviations.
Sum of squares of deviations = 0.5218 + 2.5952 + 0.0732 + 0.8493 + 0.000041 + 0.0105 + 0.0939 + 0.1074 + 0.457 + 1.712 = 6.31975
5. Calculate the variance of the independent variable (peak area).
Variance of peak area = sum of squares of deviations of peak area / (n - 1), where n is the number of data points.
Variance of peak area = 6.31975 / (5 - 1) = 1.5799375
6. Calculate the covariance between the independent variable (peak area) and the dependent variable (mole percent isooctane).
Covariance = sum of (deviations of peak area * deviations of mole percent isooctane)
Covariance = (-0.7216 * -1.612) + (-0.2706 * -0.922) + (0.0064 * -0.102) + (0.3064 * 0.328) + (0.6764 * 1.308) = 1.90836
7. Calculate the slope (m) and intercept (b) of the line.
Slope (m) = covariance / variance of peak area = 1.90836 / 1.5799375 = 1.2072
Intercept (b) = average mole percent isooctane - (slope * average peak area) = 2.702 - (1.2072 * 1.0736) = 1.39856
8. Calculate the uncertainties in the slope and intercept.
Uncertainty in slope = sqrt(variance of peak area / (n * (n - 2)))
Uncertainty in slope = sqrt(1.5799375 / (5 * (5 - 2))) = 0.64211
Uncertainty in intercept = uncertainty in slope * sqrt(1 / n + (average peak area^2 / (n * variance of peak area)))
Uncertainty in intercept = 0.64211 * sqrt(1 / 5 + (1.0736^2 / (5 * 1.5799375))) = 0.2472
9. The equation of the line (y = mx + b) is:
mole percent isooctane = 1.2072 * peak area + 1.39856
Now, we can move on to part (ii) to calculate the mole percent of isooctane in the unknown sample (peak area = 2.65) and the associated error and confidence interval.
(a) If the peak area was the result of a single measurement, you can substitute the peak area into the equation of the line to find the corresponding mole percent isooctane.
mole percent isooctane = 1.2072 * 2.65 + 1.39856 = 4.39108
To calculate the error of the mole percent, you need the standard deviation of the residuals (variation in the data points around the line of best fit). First, find the sum of the squares of the residuals:
Residuals:
(mole percent isooctane - y) = (1.09 - (1.2072 * 0.352 + 1.39856))^2
(1.78 - (1.2072 * 0.803 + 1.39856))^2
(2.60 - (1.2072 * 1.08 + 1.39856))^2
(3.03 - (1.2072 * 1.38 + 1.39856))^2
(4.01 - (1.2072 * 1.75 + 1.39856))^2
Sum of squares of residuals = the sum of all the calculated squares of residuals
The standard deviation of the residuals can be calculated as the square root of the mean of the squares of the residuals divided by (n - 2):
Standard deviation of residuals = sqrt(Sum of squares of residuals / (n - 2))
Finally, the error of the mole percent can be calculated as the standard deviation of the residuals multiplied by the slope:
Error of mole percent = Standard deviation of residuals * slope = (calculated value)
(b) If the peak area was the mean of four measurements, you would first calculate the mean peak area and then substitute it into the equation of the line to find the corresponding mole percent isooctane.
(c) To comment on the difference in answers between (a) and (b), you will compare the error and confidence intervals obtained from both cases.