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12.5: Chemical Energy Conversion

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    19017
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    Batteries and fuel cells store energy in the chemical bonds of atoms. These devices were studied in Chapter 9. Table \(\PageIndex{1}\) details how to describe the physics of these chemical energy storage devices using the language of calculus of variations.

    Sometimes chemists discuss macroscopic systems and describe charge distribution in a material by charge density \(\rho_{ch}\) in units \(\frac{C}{m^3}\). In other cases, chemists study microscopic systems, where they are more interested in the number of electrons \(N\) and the distribution of these electrons around an atom. The second and third columns of Table \(\PageIndex{1}\) specify how to describe the macroscopic systems in the language of calculus of variations while the last two columns specify how to describe the microscopic systems.

    In the second column of Table \(\PageIndex{1}\), the generalized path is \(\rho_{ch}\) and the generalized potential is the redox potential \(V_{rp}\) in volts. There is a close relationship between the choice of variables specified in the second column of Table \(\PageIndex{1}\) and the choices specified in the second columns of Table 12.2.1 and 12.2.3. More specifically, the generalized path described in the second column of Table 12.2.1 is charge \(Q\) in coulombs, where charge is the integral of the charge density with respect to volume.

    \[Q=\int \rho_{c h} d \mathbb{V} \nonumber \]

    The generalized path described in the second column of Table 12.2.3 is displacement flux density \(\overrightarrow{D}\) in units \(\frac{C}{m^2}\). In the third column of Table \(\PageIndex{1}\), the opposite choice is made with \(V_{rp}\) for the generalized path and \(\rho_{ch}\) for the generalized potential. In Chapter 13, we consider a calculus of variations problem with this choice of variables in more detail to solve for the electron density around an atom.

    Another way to apply the language of calculus of variations to chemical energy storage systems is to choose the number of electrons \(N\) as the generalized path and the chemical potential \(\mu_{chem}\) as the generalized potential [172]. This situation is described in the fourth column of the Table \(\PageIndex{1}\). We could instead choose \(\mu_{chem}\) as the generalized path and \(N\) as the generalized potential, and this situation is detailed in the last column of Table \(\PageIndex{1}\). Reference [172] details using calculus of variations with this choice of variables. Chemical potential is also known as the Fermi energy at \(T = 0\) K, and it was discussed in Sections 6.2 and 9.2.3. It represents the average between the highest occupied and lowest unoccupied energy levels. The quantity \(E_g\), which shows up in the fourth row of the table, is the energy gap in joules.

    Table \(\PageIndex{1}\): Describing chemical systems in the language of calculus of variations.
    Energy storage device Battery, fuel cell Battery, fuel cell Battery, fuel cell, chemical bonds of an atom Battery, fuel cell, chemical bonds of an atom
    Generalized Path Charge density \(\rho_{ch}\) in \(\frac{C}{m^3}\) Redox potential (voltage) \(V_{rp}\) in volts Number of electrons \(N\) Chemical potential \(\mu_{chem}\) in \(\frac{J}{\text{atom}}\)
    Generalized Potential Redox potential (voltage) \(V_{rp}\) in volts Charge density \(\rho_{ch}\) in \(\frac{C}{m^3}\) Chemical potential \(\mu_{chem}\) in \(\frac{J}{\text{atom}}\) Number of electrons \(N\)
    Generalized Capacity Capacitance \(C\) in farads \(\frac{1}{C}\) Inverse of energy gap \(\frac{1}{E_g} = \frac{\partial N}{\partial \mu_{chem}}\) Energy gap \(E_g = \frac{\partial \mu_{chem}} {\partial N}\) in J
    Constitutive relationship \(\int \rho_{c h} d \mathbb{V} = CV_{rp}\) \(V_{rp} = \frac{1}{C} \int \rho_{c h} d \mathbb{V} \) \(\Delta N = \frac{1}{E_{g}} \Delta \mu_{c h e m}\) \(\Delta \mu_{c h e m} = E_{g} \Delta N\)
    Energy \(\int_{\mathbb{V}} \rho_{c h} V_{r p} d \mathbb{V}\) \(\int_{\mathbb{V}} \rho_{c h} V_{r p} d \mathbb{V}\) \(N \mu_{chem}\) \(N \mu_{chem}\)
    Law for potential Nernst eq. (KVL) Conservation of Charge Nernst eq. (KVL) Conservation of Charge
    This column assumes no variation in \(\theta\) or \(\phi\) no variation in \(\theta\) or \(\phi\) no variation in \(\theta\) or \(\phi\) no variation in \(\theta\) or \(\phi\)

    This page titled 12.5: Chemical Energy Conversion is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.