# Classical Thermodynamic Analysis of Deuterium-Based Fusion Reactions

^{1}

^{2}

^{*}

## Abstract

**:**

^{6}–10

^{7}K) and where ΔS is negative, the change in Gibbs free energy may be positive, making the reaction non-spontaneous. This paper proposes a classical thermodynamic analysis of D-based reactions of interest for the magnetic-confinement fusion applications. The entropy contribution was evaluated via the Sackur–Tetrode equation while the change in enthalpy was considered constant and as corresponding to the Q-value of the fusion reaction. The results of the thermodynamic analysis are compared with nuclear reaction feasibility criteria based on the reaction reactivity. The DT and D

^{3}He reactions show a high degree of spontaneity although the second one presents a lower reactivity. An increase in temperature could enhance the reactivity of the D

^{3}He reaction at the cost of decreasing its thermodynamic spontaneity. Both branches of the DD reaction are characterized by a much lower thermodynamic spontaneity than that of the DT and D

^{3}He reactions. Furthermore, at the temperature of their maximum cross section, the DD reactions exhibit a largely positive change in Gibbs free energy and, therefore, are not spontaneous. At the temperature of magnetic-confinement fusion machines (1.5 × 10

^{8}K), among the D-based reactions studied, the DT one exhibits the highest degrees of spontaneity and reactivity.

## 1. Introduction

_{2}and other greenhouse gases (GHG) the national and worldwide energy strategies are aimed at advancing the penetration of renewable sources (solar, photovoltaic, biomass) and the use of cleaner energy vectors (namely hydrogen) [2,3,4,5,6].

_{p}), the temperature (T) and the energy confinement time (τ

_{E}). Their product is called the “triple product”, a parameter used to rank the performance of the tokamaks existing in the laboratories or designed for future research. Based on the balance between the thermonuclear power generated (A) and the power lost from the plasma (B), Lawson’s criterion has been introduced [16]. This criterion has been given in several versions (e.g., “plasma breakeven”, “ignition”) that differ each other for the fraction of B considered to be useful for providing energy to the plasma (e.g., that carried by alpha particles) or that may be converted to electricity. For instance, for the DT reaction, under the assumption that the fusion power carried out by the alpha particles is equal to the losses for conduction, Lawson’s criterion leads to calculating the following triple product:

_{p}T τ

_{E}≥ 3 × 10

^{21}m

^{−3}keV s

^{4}He and a neutron. This reaction exhibits a high value of maximum cross section at a relatively low temperature:

^{3}Helium, then, is:

_{B}= 1.16 × 10

^{7}K, where k

_{B}is the Boltzmann constant).

^{3}/s. The reaction rate, i.e., the number of reactions per unit time and volume, is then obtained from the reactivity via its product with the density of the particles involved. Once the distribution of the velocities of the reacting nuclei is defined, it is possible to evaluate an average reactivity <σ v> through the parameterization of experimental data [17]. For the scope of this work, the average reactivities of the D-based fusion reactions, shown in Figure 1, have been calculated by considering Maxwellian velocity distributions and using simplified formulas from the literature [18,19].

## 2. Process Analysis via Classical Thermodynamics

^{−1}), is expressed by:

^{−1}), T (K) the temperature and S the entropy (J mol

^{−1}K

^{−1}). The change in G can be calculated at constant temperature and pressure by:

^{6}–10

^{7}K) and where ΔS is negative, the change in Gibbs free energy may be positive (i.e., when |T ΔS| > |ΔH|) making the reaction non-spontaneous above the temperature at which the condition ΔG = 0 occurs. Such a temperature, that establishes the achievement of the equilibrium conditions and then the passage from a spontaneous (ΔG < 0) to a non-spontaneous (ΔG > 0) reaction, was introduced and defined as follows under the hypothesis of constant pressure and temperature [20]:

^{int}the partition function of each particle (or molecule). For monoatomic compounds f

^{int}is equal to 1.

^{23}particles), P is the pressure (atm), M the molecular weight and R the gas constant.

^{4}He, tritium and

^{3}He) plus an amount of heat corresponding to the Q-value. Such an assumption complies with the architecture of the tokamak machines presently studied where the energy carried by the neutrons is changed into heat at the level of the shielding and/or blanket systems.

^{−5}atm (≈5 Pa), a value in agreement with both experiments and designs of magnetic fusion devices [24,25]. According to Formula (11), the change of pressure has a modest influence on the entropy assessment and, therefore, the variation of this parameter has not been considered.

_{700 K}− [T S]

_{T*}

- -
- the reacting particles, at plasma state, behave like a perfect gas;
- -
- the change in enthalpy is assumed to be constant and is defined by Formula (7);
- -
- according to the initial and final states defined for fusion reactions (12–15), the contribution of the sub-nuclear particles (neutrons, protons, etc.) to the assessment of the thermodynamic functions is neglected.

## 3. Results and Discussion

^{12}J/mol), and, therefore, these reactions are very spontaneous from the thermodynamic point of view. Both branches of the DD reaction, formulas (2) and (3), show quite similar values of ΔG along the T as well: these values at low temperature are about minus 3.2 and 4.0 MeV (i.e., minus 3.1–3.9 × 10

^{11}J/mol), indicating a spontaneity level 4–5 times lower than those of reactions (1) and (4).

^{8}K (≈13 keV). At this temperature the reactivity <σ v> of the reaction (1) is 1.91 × 10

^{−22}m

^{3}/s and the ΔG is about −16.0 MeV. These values indicate a high degree of thermodynamic spontaneity of the DT reaction when carried out in the magnetic-confinement devices existing or under design.

^{8}K a high spontaneity level (ΔG = −16.8 MeV), it appears to be less practicable since its reactivity is about 7.40 × 10

^{−25}m

^{3}/s, i.e., three orders of magnitude lower than that of reaction (1). On the contrary, at a temperature of 2.9 × 10

^{9}K (around 250 keV), corresponding to its maximum cross section, the ΔG reaction is positive (9.81 MeV), showing that the process is not spontaneous. This is a clear example of conflict between thermodynamic and kinetic optimization: best values of ΔG occur at temperature (1.5 × 10

^{8}K) quite below its maximum reactivity <σ v> that instead takes place at 2.9 × 10

^{9}K. In other words, by operating the reaction (4) at temperatures higher than 1.5 × 10

^{8}K the thermodynamics foresees a reduction of the amount of D and

^{3}He converted although its reaction probability increases.

^{8}K, their thermodynamic spontaneity is, respectively, verified (ΔG < 0) although at these temperatures their reactivity is very poor. For instance, at 1.5 × 10

^{8}K, ΔG is −1.77 and −2.53 MeV for reactions (2) and (3), respectively. At this temperature, the reactivity of these reactions is about 10

^{−24}m

^{3}/s against the value of 10

^{−22}m

^{3}/s exhibited by reaction (1), as verified in fusion experimental devices where the amount of DD fuel reacting is at least two orders of magnitude lower than that of the DT reaction.

## 4. Conclusions

^{8}K), both reactions DT and D

^{3}He exhibit a high level of spontaneity, although the last one presents a lower reactivity. An increase in temperature could enhance the reactivity of reaction D

^{3}He despite, however, decreasing its thermodynamic spontaneity since it results in a ΔG = 0 at 1.65 × 10

^{9}K.

^{3}He reactions. In particular, at the temperature of their maximum cross section where the reaction kinetics is favoured, the DD reactions exhibit a largely positive change in Gibbs free energy since their entropic contribution (TΔS) is not balanced by the Q-value (−ΔH).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Change of Gibbs free energy of the D-based reactions reported in the graph legend as follows: (1) = DT, (2) = DD -> T, (3) DD ->

^{3}He and (4) = D

^{3}He.

T* keV (K) | ΔG at 13 keV (1.5 × 10 ^{8} K)MeV | <σ v> at 13 keV (1.5 × 10^{8} K)m ^{3}/s | T of σ_{max}keV (K) | ΔG (at T of σ_{max})MeV | |
---|---|---|---|---|---|

DT | 1.32 × 10^{2}(1.58 × 10 ^{9}) | −16.0 | 1.91 × 10^{−22} | 64 (7.42 × 10 ^{8}) | −9.51 |

DD -> ^{3}He | 2.67 × 10^{1}(3.10 × 10 ^{8}) | −1.77 | ≈1 × 10^{−24} | 1750 (2.03 × 10 ^{10}) | 2.43 × 10^{2} |

DD -> T | 3.34 × 10^{1}(3.88 × 10 ^{8}) | −2.53 | ≈1 × 10^{−24} | 1250 (1.45 × 10 ^{10}) | 1.70 × 10^{2} |

D^{3}He | 1.42 × 10^{2}(1.65 × 10 ^{9}) | −16.8 | 7.40 × 10^{−25} | 250 (2.90 × 10 ^{9}) | 9.81 |

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Tosti, S.; Marrelli, L.
Classical Thermodynamic Analysis of Deuterium-Based Fusion Reactions. *Hydrogen* **2022**, *3*, 53-61.
https://doi.org/10.3390/hydrogen3010004

**AMA Style**

Tosti S, Marrelli L.
Classical Thermodynamic Analysis of Deuterium-Based Fusion Reactions. *Hydrogen*. 2022; 3(1):53-61.
https://doi.org/10.3390/hydrogen3010004

**Chicago/Turabian Style**

Tosti, Silvano, and Luigi Marrelli.
2022. "Classical Thermodynamic Analysis of Deuterium-Based Fusion Reactions" *Hydrogen* 3, no. 1: 53-61.
https://doi.org/10.3390/hydrogen3010004