## 1 Introduction
### Algorithms for Optical Particle Sensors  
#### 1.2.1 Standard Algorithm for Optical Particle Sensors  
The standard algorithm employed by many manufacturers of optical particle counters since the 1970s is a completely open and transparent approach. It typically uses three bins (0.3-0.5 \(\upmu\)m, 0.5-1 \(\upmu\)m, and 1-2.5 \(\upmu\)m) to calculate PM\({}_{2.5}\). For each bin, the assumption is made that all particles are spherical and have identical diameters \(\mathrm{D_{P}}\) equal to some midpoint (either arithmetic or geometric mean) between the bin boundaries. The volume of the single particle is then \(\pi\mathrm{D_{P}}^{3}/6\). All particles are assumed to have the same density \(\rho\). The mass of the particle is then equal to the density multiplied by the volume: \(\rho\pi\mathrm{D_{P}}^{3}/6\). The total mass of particles in the bin is equal to the single-particle mass multiplied by the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Instrument Tested by AQ-SPEC in Laboratory Report** & **Plantower Model** \\ \hline Airbeam 2 & 7003 \\ \hline Airbeam 3 & 7003 \\ \hline Air Quality Egg & 5003 \\ \hline APT Minima & A003 \\ \hline AS-Lung & 3003 \\ \hline Davis Instruments & A003 \\ \hline PurpleAir PM-I & 1003 \\ \hline PurpleAir PM-II & 5003 \\ \hline PurpleAir Flex & 6003 \\ \hline Redspira & 5003 \\ \hline Sain Smart & 5003 \\ \hline Smart Citizen Kit & 5003 \\ \hline Magna SCI SRL uRAD SMOGGE & A003 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Low-cost particle instruments using Plantower sensors tested in chamber studies by AQ-SPEC.  
number N\({}_{\text{i}}\) of particles in the bin. Finally, PM\({}_{2.5}\) is equal to the sum of the masses of all particles in the three bins:  
\[\text{PM}_{2.5}=\text{aN1}+\text{bN2}+\text{cN3} \tag{1}\]  
where a, b, and c are simply the masses of the single (representative) particle in each bin and N1, N2, and N3 are the number of particles in each bin. Numerically, assuming the choice of geometric mean for the particle diameter and a density of 1 g cm\({}^{-3}\), the values of a, b, and c are as follows (Table 2):  
In this table, the volumes and masses have been divided by 100 since the number N\({}_{\text{i}}\) of particles is in units of number per deciliter. This allows one to use the numbers reported by Plantower for N1, N2, and N3 without change.  
Therefore, the alt PM\({}_{2.5}\) algorithm for a density of 1 g cm\({}^{-3}\) is given by Equation (1) using the values of a, b, and c shown in the above table.  
As with all uses of PM\({}_{2.5}\) estimates, however, it is always recommended that investigators compare the PM\({}_{2.5}\) predictions to research-grade instruments measuring the aerosol mixture of interest. In the field, this is usually done by comparing nearby regulatory monitors using gravimetric Federal Reference Method or continuous Federal Equivalence Method (FEM/FRM) to the aerosol mixture being measured. The result of the comparison is a calibration factor (CF) to adjust the PM\({}_{2.5}\) estimates. For the choice of geometric mean and density of 1 as mentioned above, the final equation for PM\({}_{2.5}\) becomes  
\[\text{PM}_{2.5}=\text{CF}(\text{aN1}+\text{bN2}+\text{cN3}) \tag{2}\]