## 3 Problem Statement
### Relevant Physical and Chemical Principles and Laws  
In our use case, we have sensor data for the temperature and pressure measurement. To formalize meaningful features enabling the models to predict C5 content, we have considered the following laws and equations from physics:  
* **[PERSON]'s law** states that the total pressure of a component equals the vapor pressure of its pure components multiplied by their mole fraction (see Equation (1));
* **[PERSON]'s equations** provide a relationship between the vapor pressure of a pure component and three empirically measured constants at a given temperature (see Equation (2));
* **Combined Gas Law** states that the ratio of the product of pressure and volume and the absolute temperature of a gas equal a constant (see Equation (3));
* **Clausius-Clapeyron relation** describes pressure at a given temperature T\({}_{2}\) if the enthalpy of vaporization and vapor pressure are known at some other temperature T\({}_{1}\) (see Equation (4)) \[P=P_{1}\cdot x_{1}+\ldots+P_{n}\cdot x_{n}\] (1)  
Equation (1): [PERSON]'s law. \(P\) refers to pressure, \(x\) refers to mole fraction, and the \(n\) indicates different mixture components. \[log_{10}P=\frac{A^{\
eg}B}{C+T} \tag{2}\]  
Equation (2): [PERSON]'s equation. \(P\) refers to pressure, \(T\) refers to temperature. \(A\), \(B\), \(C\) are empirical, component specific constants. \[k=\frac{P\cdot V}{T} \tag{3}\]  
Equation (3): Combined Gas Law equation. \(P\) refers to pressure, \(V\) refers to volume, and \(T\) refers to temperature. \(k\) is a constant. \[ln\left(\frac{P_{1}}{P_{2}}\right)=-\frac{L}{R}\cdot\left(\frac{1}{T_{2}}- \frac{1}{T_{1}}\right) \tag{4}\]  
Equation (4): Clausius-Clapeyron relation. \(P\) refers to pressure, \(T\) refers to temperature, \(L\) is the specific latent heat of the substance, and \(R\) is the specific gas constant. For the case study, we obtained data from sensors P1 and T2 of the debutanizer unit (in the Figure 1); while we missed sensor readings from T1 and P2. Not having both temperature and pressure at a given point of the debutanizer column prevents us from using the Ideal Gas Law equation to compute the gas molar weight (see Equation (5)) of the mixture. The gas molar weight could provide further insights on the mixture composition using the gas molar weight equation (see Equation (6)). The gas molar weight equation expresses that the gas molar weight equals to the sum of the molar weights of the pure components multiplied by their mole fraction. \[P\cdot V=n\cdot R\cdot T \tag{5}\]  
Equation (5): Ideal Gas Law. \(P\) stands for pressure, \(V\) stands for volume, \(n\) represents the amount of substance, \(R\) is the ideal gas constant, and \(T\) corresponds to the temperature. \[M=M_{1}\cdot x_{1}+\ldots+M_{n}\cdot x_{n} \tag{6}\]  
Equation (6): Molar weight equation. \(M\) stands for molar weight, \(x\) represents mole fractions, while the subindexes indicate different mixture components. LPG specifications require LPG to have a mixture of propane and butane, with no more than 2% of the volume of five carbon components (C5) and no more than 5% of the volume of two and five carbon components (C2 and C5). Though many possible components have two and five carbons, we decided to approximate them as a single pure component. Considering the laws, equations, and restrictions described above, we can derive a set of equations, which provide meaningful cues on the expected mixture composition, and thus drive better forecasts. e.g., from Equation (1) and approximating the LPG composition to the four elements described above, we obtain that C5 proportion can be expressed as Equation (7). Although pressure is known from the sensor readings (P1), we do not know the exact proportion of propane and butane. We also miss sensor data regarding the temperature at the same point where the pressure is sensed (T1). Considering that the relationship between temperature and pressure is linear, and given a snapshot of sensor data, we approximate T1 based on P1 and T2.