# Development of a New Evaporation Exposure Model: Chemical Product Evaporation Model (CPEM)
## 2 Theory  
### Derivation of the CPEM  
Here, we develop the CPEM using the mass balance of substances that evaporate from a product into the air:  
\[\frac{dM_{air}}{dt}=\dot{J}-QC_{air}, \tag{1}\]  
where \(M_{air}\) is the mass of the target substance in the air ([g]), \(\dot{J}\) is the flux of a substance into the air through surface contact between the air and product ([g/h]), \(Q\) is the air flow rate by ventilation ([m\({}^{3}\)/h]), and \(C_{air}\) is the substance concentration in the air ([g/m\({}^{3}\)]). According to [3], \(\dot{J}\) can be described by [PERSON]'s first law as follows:  
\[\dot{J}=-\frac{dM_{prod}}{dt}=-K\mathcal{S}\Delta C_{air}=-\frac{KSM}{RT}\big{(} P_{eq}-P_{air}\big{)}, \tag{2}\]  
where \(M_{prod}\) is the mass of a substance in the product ([g]), \(K\) is the mass transfer rate ([m/h]), \(S\) is the surface area of the product ([m\({}^{2}\)]), \(M\) is the molecular weight of the substance ([g/mol]), \(R\) is the universal gas constant, \(T\) is the temperature ([K]), \(P_{eq}\) is the equilibrium vapor pressure ([Pa]), and \(P_{air}\) is the actual vapor pressure. From Equations (1) and (2), the derivative for the concentration in the air is derived as follows:  
\[\frac{dC_{air}}{dt}=\frac{KSM}{VRT}\big{(}P_{eq}-P_{air}\big{)}-\frac{Q}{V}C_{ air}, \tag{3}\]  
where \(V\) is the room volume ([m\({}^{3}\)]), and \(Q\) is the ventilation rate ([m\({}^{3}\)/h). According to [PERSON]'s law and the ideal gas equation, Equation (3) can be redescribed as follows:  
\[\frac{dC_{air}}{dt}=\frac{KSM}{VRT}\left(\frac{P_{vap}C_{prod}}{C_{prod}+C_{r} \frac{M}{M_{r}}}-\frac{RT}{M}C_{air}\right)-qC_{air}, \tag{4}\]  
where \(C_{prod}\) is the concentration of target substance in a product ([g/kg]), \(C_{r}\) is the concentration of the other materials in the product ([g/kg]) and is assumed to be constant, \(P_{vap}\) is the saturated vapor pressure of the target substance, \(M_{r}\) is the average molecular weight of the other materials ([g/mol]), and \(q(=Q/V)\) is the ventilation rate coefficient ([h\({}^{-1}\)]). The flux for the other materials can be expressed according to [PERSON]'s first law as follows:  
\[\dot{j}^{r}=KSAC_{air}^{r}=\frac{KSM^{r}}{RT}\Big{(}P_{eq}^{r}-P_{air}^{r} \Big{)}\cong\frac{KSM^{r}}{RT}\Big{(}P_{vap}^{r}-P_{air}^{r}\Big{)}, \tag{5}\]  
where \(\dot{j}^{r}\) is the mass flux of the other materials per unit time through surface contact between the air and product, \(P_{eq}^{r}\) is the equilibrium vapor pressure of the other materials, \(P_{air}^{r}\) is the actual vapor pressure of the other materials, and \(P_{vap}^{r}\) is the saturated vapor pressure of the other materials. If the other materials accounted for most of the product, \(C_{prod}\) was negligible, so \(P_{eq}^{r}\) was almost equal to \(P_{vap}^{r}\). Because \(dM_{air}^{r}/dt=\dot{j}_{p}^{r}-QC_{air}^{r}\), dividing by \(V\) gives the following:  
\[\frac{dC_{air}^{r}}{dt}=\frac{KSM^{r}}{VRT}P_{vap}^{r}-\left(\frac{KS}{V}+q \right)C_{air}^{r}, \tag{6}\]  
where \(C_{air}^{r}\) is the concentration of other materials in air ([g/m\({}^{3}\)]). Because all the parameters of Equation (6) are constants, \(C_{air}^{r}\) is stable on a daily basis if the room air is ventilated once per day. After each active ventilation, the same flux is repeatedly generated from the product surface area because the concentration of other materials in the product does not change. Thus, the product mass reduction rate can be considered constant. This means that the change in product mass can be described by zero-order kinetics as follows:  
\[\frac{dA_{prod}}{dt}=-\mu, \tag{7}\]  
where \(A_{prod}\) is the product mass ([kg]) and \(\mu\) is the decreasing rate of product mass ([kg/hr]).