stylistic:style_transfer, linguistic:paraphrasing
Thermodynamics of bubbles

The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations. The above free energy can be written as: Ω = − Δ P V A − p B V + γ A {\displaystyle \Omega =-\Delta PV_{\text{A}}-p_{\text{B}}V+\gamma A} where Δ P = p A − p B {\displaystyle \Delta P=p_{\text{A}}-p_{\text{B}}} is the pressure difference between the inside (A) and outside (B) of the bubble, and V A {\displaystyle V_{\text{A}}} is the bubble volume. In equilibrium, d Ω = 0, and so, Δ P d V A = γ d A. {\displaystyle \Delta P\,dV_{\text{A}}=\gamma \,dA.}

For a spherical bubble, the volume and surface area are given simply by V A = 4 3 π R 3 → d V A = 4 π R 2 d R, {\displaystyle V_{\text{A}}={\tfrac {4}{3}}\pi R^{3}\quad \rightarrow \quad dV_{\text{A}}=4\pi R^{2}\,dR,} and A = 4 π R 2 → d A = 8 π R d R. {\displaystyle A=4\pi R^{2}\quad \rightarrow \quad dA=8\pi R\,dR.}

Substituting these relations into the previous expression, we find Δ P = 2 R γ, {\displaystyle \Delta P={\frac {2}{R}}\gamma,} which is equivalent to the Young–Laplace equation when R = R.