Final answer:
The energy change during the coalescence of two spherical nuclei is calculated by analyzing the change in surface energy due to surface tension and is found to be negative, indicating an energy reduction.
Step-by-step explanation:
To calculate the energy change during the coalescence of two spherical nuclei, we consider surface tension and the free energy change per unit volume. The surface area of the initial separate spheres is given by ΣA₁ = 4πr₁² for the first nucleus and ΣA₂ = 4πr₂² for the second nucleus. After coalescence, we have a single spherical nucleus with radius r given by r³ = r₁³ + r₂³ due to volume conservation. Its surface area is ΣA = 4πr².
The surface energy before coalescence is ΣE₁ = ΣA₁∙γ + ΣA₂∙γ, and after coalescence, it is ΣE = ΣA∙γ. The change in surface energy is ΔΣE = ΣE - ΣE₁. Since a sphere minimizes surface area for a given volume, ΔΣE is negative, indicating an energy reduction due to lower surface area.
The free energy change due to volume change is ΔG = ΔGᵢ∙ΣV, where ΣV is the total volume of the new nucleus. Because ΣV is retained during coalescence, this term does not contribute to the energy change calculation for the coalescence process.
Therefore, the total energy change during coalescence, in terms of surface tension γ and free energy change per unit volume ΔGᵢ, is primarily the change in surface energy, which will be negative as required.