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In this paper we present the analogous electrical model for analyzing and determining the precise time dependence of concentrations in general first and zero order chemical reactions. In addition, the applicability of this analogous electrical model for investigating the optical and bio chemical processes is also presented. By constructing the proper analogous electrical circuit experimentally or with the help of special electrical software, the time behavior of the analyzed parameter even for extremely complicated processes can be obtained.

The kinetics of chemical reactions is usually described by a set of differential equations where the rate of the reaction is a function of concentration. The general form is defined by [

In this paper, we present how a voltage can be used as analogous, among other for analyzing the concentration of chemical kinetic reactions either by building a compatible electrical circuit and measuring its output voltage time behavior or by using special electrical software [

In addition, the analogous electrical model can be adapted also to other fields such as the rate equation of atoms concentrations in laser as well as to investigate the time dependence of concentration in biological and biochemical problems which are usually very difficult issue to handle. In order to solve those problems a steady state assumption must be used [

Moreover, If a scientist would like to investigate a physical or chemical processes whose parameters are unknown, he can build an analogous electrical circuit and study the output of this circuit by changing the value of one of its components (as a resistor). The output voltage (which is analogous to the examined parameter) will be obtained as fast as the speed of light. On the other hand, the scientist may use the “hardware solution” of the analogous electrical model in order to either learn about the behavior of differential equations having one or more symbolic parameters (

In this section, we present the analogous quantities of the electrical model related to the chemical kinetic equations. The solution of the electrical circuit is achieved either by proper electrical simulation software or by building the circuit experimentally and measuring its output. This procedure provides a precise solution of the material concentrations as a function of time in the examined chemical reaction. The proposed analogous model, described by Equations (2–6) defines the chemical components (_{0} ⇔ _{0}_{0} is the volume of the reaction,

At first, we should examine the consistency of this analogous electrical model where:

By applying the analogous model one may derive:

And this corresponds to the basic relation of a capacitor. Another example for the consistency of this model is:

This relation equals to the definition of an electrical current. By applying the model one obtains that:

Since _{0} is a constant, this relation can be rewritten as:

Indeed this is the definition of the reaction’s velocity. Thus, it may be concluded that chemical kinetic reactions can be represented by a simple capacitor–resistor circuit, which is very simple to solve. By measuring or calculating a certain voltage over an electrical parameter, one may know the behavior of the materials concentration as a function of time at the required examined chemical component.

For each chemical element a different analogous circuit should be drawn.

The circuit is drawn only according to the chemical reaction equations.

First a capacitor is drawn. The voltage upon this capacitor (_{C}) will be analogous to the concentration of the certain element.

Each arrow getting out of this element in its chemical reaction equation will be represented by a resistance in parallel to the capacitor. This is because each resistance is drawing current from the capacitor, an effect that is analogous to the “chemical current drawing”,

Each arrow pointing toward our element will correspond to a controlled current source, parallel to the resistance and the capacitor. Each current source like this will provide current to the capacitor, a phenomenon that is analogous to “chemical reaction” supplied by the decomposing element for the creation of a new element. The value of the current will be _{i}V_{i}

A sketch of the most general first order chemical kinetic reaction.

In _{1}, while _{2} and X_{3}, respectively. Please note, that the symbol of the current source represents a controlled current source. This current source is proportional to a voltage applied at different branch along the circuit.

The analogous electrical circuits correspond to the most general first order reaction. (_{1}. (_{2} and X_{3}, respectively.

In order to substantiate the theory, we examined the applicability of the proposed method in the most common chemical reactions based on the analogous relations presented by Equations (2–6).

The analogous electrical circuit for zero order reactions.

The electrical solution for this circuit is:

Thus
_{out}(_{c,o} is the initial voltage on the capacitor. After applying the analogous model, _{out}(_{c,0} corresponds to the initial concentration of the decomposing material, thus:
_{0} –

According to the dimensional analysis of the analogous model, the dimensions of

In

The analogous electrical circuit for the first order reactions represented by a resistor-capacitor circuit.

The solution of this electrical circuit is:

Which is exactly the solution for the first order reaction that equals to:
_{0}^{−kt}

The analogous electrical circuit for reversible reactions.

Where, _{in} is analogous to the initial concentrations of A and B.
_{in} ⇔ [_{0} + [_{0}

The differential equation describes the electrical circuit of

Using the analogous table one may see that:

Thus, Equation (18) is exactly identical to the differential equation of the reversible chemical reaction and therefore the analogous solution is identical as well. Thus the solution is:

Therefore, the equilibrium constant in this case is equal to:

From Equations (21) and (22) for

Thus, the equilibrium constant converges to the analogous constant of the electric circuit. The voltages of _{B} and _{C} represent the analogous time behavior for the concentration of material B ([_{b}→ ∞ (_{b} = 0, the reaction becomes

A triple consecutive reaction described by _{b} = 0.1 [min^{−1}], _{c} = 0.05 [min^{−1}], [_{0} = 1, [_{0} = 0, [_{0} = 0. _{A}, _{B} and _{C} are the analogous voltages of concentrations [

The analogous electrical circuit describing a triple consecutive reaction of concentrations [

The output voltage (

In [

Vapor phase decomposition of ethylene oxide into methane and carbon monoxide.

The given data is related to the total pressure dependence upon time and since the reaction takes place in a vapor phase one may apply the ideal gas law:
_{4} and CO (the voltages that correspond to CH_{4} and CO elements) were added. The obtained total pressure is:
_{0}^{−kt}_{0}(1 − ^{−kt}) = _{0}(2 − ^{−kt})

The rate constant ^{−1}], which is indeed the correct rate constant of this reaction.

Vapor phase decomposition of ethylene oxide. A comparison between experimental data and simulation results acquired using the analogous electrical model.

The analogous electrical model for measuring and calculating the precise time behavior of concentration in general chemical reaction has been shown in

The rate of the induced emission or absorption influenced by the power of the radiation^{1}^{0} can be described by:
_{m}_{sp} is the spontaneous lift time of the level, _{o}) is the chart of the atoms level spectral reaction, ρ(υ) is the spectral structure of the illuminating radiation having units of _{o} is the atom’s resonant frequency. h = 6.626 × 10^{−34} [J·sec] is the Planck’s constant. The common case is related to a narrow band illumination described by:
_{υ} δ_{i} will be:

In the investigated example, a pressure broadening is assumed and thus _{o}) is:
_{o} and thus:

This model is applied on a four level laser configuration as illustrated in

The energy model of a four level laser.

Where _{32} = υ_{3} − υ_{2} and thus the induced emission and absorption occurs between the second and the third energy levels. The rate equations^{9} are as follows:
_{i} and τ_{spi} are the concentration of the atoms and the relaxation time related to the spontaneous process at the _{0} + _{1} + _{2} + _{3}

The analogous electrical model of atoms concentrations at each one of the four levels of the laser.

In this section, we analyzed the time behavior of concentrations flowing through a membrane. The definition of flux, _{0}

The flux coming from diffusion:

We will assume that the concentration of the material is changed linearly between the two sides of the membrane as illustrated in

Schematic sketch of the concentration distribution assumption.

The analyzed case is a membrane separating between two cells, whereas each cell contains different concentrations of NaCl. We will denote the concentration of Na and Cl in the left cell as [Na^{+}]_{1} and [Cl^{−}]_{1}, respectively and [Na^{+}]_{2} and [Cl^{−}]_{2} for the right side, respectively. In our system, we chose the interesting case where only the Na can penetrate the membrane and the Cl cannot. Due to the diffusion, atoms of Na will penetrate the membrane and electrical field will develop. Eventually this field will stop the diffusion potential and the membrane will be charged with electrical charge according to the law of Gauss. Thus, producing an electrical field of:
_{0}_{0}_{w}_{w} is the molar weight of the material. The equation for the total flux is:

With our assumptions one may write the following set of equations:
_{1}_{2}

The material conservations law is:

As shown before, the flux coming from the electrical field is proportional to the multiplication between the electrical field and the concentration. The concentration of [Na^{+}]_{2} was taken since the right side of the membrane will be occupied only by Na^{+} and the left side by Cl^{−}. No Na^{+} will be near the membrane’s left side due to the rejection of the electrical field. Because of this reason, Cl^{−} appears at the membrane’s right side. Equations (47) and (48) are being easily modified according to the analogous electrical model. Thus, the analogous electrical circuits can be illustrated according to _{o}.

The analogous electrical circuits of concentrations [Na^{+}]_{1} and [Na^{+}]_{2} related to the flow through cell’s membrane.

If the membrane is made out of metal, the equations become much simpler, since no electric potential exists between the two cells:

The conservation of material:

The analogous circuit for this case is illustrated in

The analogous electrical circuit of concentrations [Na^{+}]_{1} and [Na^{+}]_{2} that are related to a flow through a metallic membrane.

In this case, we investigate an osmosis system in which the intermediate membrane is penetrable only for water. The schematic sketch of the system is illustrated in

Schematic sketch of the osmosis system. The intermediate membrane is penetrable only for water.

In the left cell, there is only water while on the right cell there is water and dissolved solid. In this system the flux is [_{2}^{w}_{2}_{w} is the specific weight of water. We will assume that between the two cells there is a linear change of the chemical potential and the pressure. Thus:
_{s}^{w}^{w} is the molar fraction of water defined by:
^{w} is the number of moles of water and ^{s} is the number of moles of the dissolved solid. Thus:

Since in the left cell (

One may write
_{2} − _{1} = _{2}^{s}_{2}^{s}

Now, we shall explore the pressure difference between the two cells:
_{2} − _{1} = _{2}_{2} − _{1}_{1})_{1}, _{2} are the heights of the water level in each cell and ρ is the specific weight. Approximately, one may write:
_{2} ≈ _{1} = _{w}
_{w} is the specific weight of water. Since:
_{1}, _{2} denote the volumes of the left and right cells, respectively. Using

The obtained heights are thus:
_{1} = _{2} =

And the flux equations are:

Those equations can convert into the electrical circuits according to the aforementioned analogous electrical model. The equivalent circuits are illustrated in

The analogous electrical circuit of number of moles

Note that the capacitor whose voltage is analogous to the concentration _{2}, while the capacitor whose voltage is analogous to the concentration _{1}.

In this paper, we have presented the analogous electrical model for analyzing and determining the time dependence of variety of applications in the fields of chemistry, biology, bio-chemistry and optics. The main advantage of this analogous model is that it makes easier to investigate chemical or physical processes whose parameters are unknown, by applying the proper analogous electrical circuit. The latter allows one to study the output of this circuit by changing the value of one of its components (