1. A student was requested to estimate the ambient concentration of water soluble sulfate in the fine particulate matters (i.e. PM2.5) at a certain location. The students obtained a filter sample of PM2.5 and then used Ion Chromatograph (IC) method to determine the concentration of the sulfate. The students made 10 repeated IC measurements (having injected the samples ten times to the IC and obtained ten chromatograms from which the sulfate concentration in PM2.5 was estimated). The results, i.e. mass concentration of sulfate in the PM2.5 sample, are tabulated below. conc, μg/m3

1 13.4

2 13.2

3 12.9

4 13.3

5 13.6

6 13.3

7 13.6

8 13.5

9 13.0

10 13.2

a. Estimate the mean conc of sulfate in the PM2.5 with the 95% confidence interval. [3 points]

b. What would be the mean conc of sulfate at 95% confident interval, if the students only made 4 replicate measurements (i.e. only measurements #1 – #4)? [3 points]

c. What is the benefit of doing replicate measurements to the estimation of the mean conc of the target species? [2 points]

d. Please give TWO practical reasons on why we usually only do a finite number of replicate measurements. [2 points]

The data given below were a lognormally distributed size of atmospheric aerosols obtained at a location A:

Size Bin, μm

ΔN, Number Conc of Particles detected in the Interval (# cm-3)

0.1-0.2

70

0.2-0.4

500

0.4-0.7

1200

0.7-1.0

1230

1.0-2.0

1830

2.0-4.0

720

4.0-7.0

112

7.0-10

10

a. Calculate the mean size for each of the size bins above for logDp distributions and tabulate the mean sizes in a table alongside the size bins. (Hints: The mean size of the size bin on a log10 scale is given by the geometric mean of the size limits.) [1 point]

b. Calculate the normalized quantities ΔN/ΔlogDp, ΔS/ΔlogDp, and ΔV/ΔlogDp, where S = surface area and V = volume and complete the size-distribution table. (Hints: S =πD2 and V =πD3/6, where D is the mean size of the particles for the specific size bin) [3 points]

c. Plot ΔN/ΔlogDp, ΔS/ΔlogDp, and ΔV/ΔlogDp as the discrete size distributions with x-axis in log DB. These are the size distribution plots that we usually see in literature and reports. [3 points]

d. Prepare the cumulative distribution table for the N distribution and calculate the corresponding cumulative percentage distribution. Remember to list the upper size limit size by size with the cumulative number concentration [3 points]

e. Plot the log-normal distributions for N concentration using the lognormal graph paper, which you may find in the notes – Ch2_StatisticalAnalysis-air pollutants. And determine the geometric mean diameter and geometric standard deviation of the distribution. (You may refer to the notes on the lognormal distribution in Statistical Analysis of Air Pollutants.) [5 points]

Note that the grade will be calculated and displayed on a 0-10 point scale.