= Solution
\b[<Availability> and total <entropy>.] The reservoir changes by $dU_R=-dU$ and $dV_R=-dV$. Its <particle number> and the system's <particle number> are fixed. The reservoir's <first law of thermodynamics> therefore gives
$$
dS_R=\frac{dU_R+p_RdV_R}{T_R}
=-\frac{dU+p_RdV}{T_R}.
$$
Add the system's <entropy> change. Because $T_R,p_R$ are constant,
$$
dS_{\mathrm{tot}}=dS-\frac{dU+p_RdV}{T_R}
=-\frac1{T_R}d(U+p_RV-T_RS).
$$
Thus define the <availability>
$$
\boxed{A=U+p_RV-T_RS,\qquad\Delta S_{\mathrm{tot}}=-\frac{\Delta A}{T_R}.}
$$
An additive constant in $A$ is immaterial; its value above the equilibrium reference is the <exergy>. The <Second law of thermodynamics> gives $\Delta S_{\mathrm{tot}}\geq0$, hence $\Delta A\leq0$. A stable equilibrium maximizes total <entropy> over the allowed variations and therefore minimizes $A$. The environmental $T_R,p_R$ enter this definition even when the system has not yet reached their values. Because both thermal and mechanical exchanges are allowed, equilibrium here includes mechanical balance; equality of temperatures alone does not ensure the minimum.
\b[<Probability> and curvature.] For a fixed value of a macroscopic coordinate $x$, let $S(x)$ count the compatible system <microstates>. The product of the system and reservoir multiplicities is
$$
\mathcal W(x)\propto
\exp\left[\frac{S(x)}{k_B}-\frac{U(x)+p_RV(x)}{k_BT_R}\right]
=\exp\left[-\frac{A(x)}{k_BT_R}\right].
$$
All factors independent of $x$ enter a normalization constant, giving
$$
\boxed{P(x)=\aleph e^{-A(x)/(k_BT_R)},\qquad
\aleph^{-1}=\int e^{-A(x)/(k_BT_R)}dx.}
$$
Here $A(x)$ is the constrained or effective <availability> for this coordinate, including the multiplicity of any eliminated variables. A nonuniform coordinate <measure> must likewise be retained or incorporated in that effective <availability>.
At a stable interior minimum, $A'(x_0)=0$ and $A''(x_0)>0$. Its <Taylor expansion> gives
$$
A(x)=A(x_0)+\frac12A''(x_0)(x-x_0)^2+\cdots.
$$
Keeping the quadratic term yields a <normal distribution>,
$$
P(x)\simeq\sqrt{\frac{A''(x_0)}{2\pi k_BT_R}}
\exp\left[-\frac{A''(x_0)(x-x_0)^2}{2k_BT_R}\right],
\qquad
\boxed{\left\langle(x-x_0)^2\right\rangle\simeq\frac{k_BT_R}{A''(x_0)}.}
$$
This is <availability curvature and equilibrium fluctuations>. The quoted fluctuation formula holds at quadratic order in the <Gaussian fluctuation approximation>; it is not an exact identity for an arbitrary nonlinear $A(x)$. Higher terms can shift the mean away from its most probable value $x_0$.
\b[Bubble equilibrium.] Keep the number of air <molecules> fixed and include the <surface energy> in the system's total <internal energy>. For constant <surface tension> $\Gamma$, this contribution is $\Gamma\Omega=\Gamma\alpha V^{2/3}$, with no surface <entropy> under the stated approximation. Its differential is
$$
dU=T\,dS-p_{\mathrm{air}}\,dV+\Gamma\,d\Omega.
$$
Hence
$$
dA=(T-T_R)dS+
\left[p_R-p_{\mathrm{air}}+\frac23\Gamma\alpha V^{-1/3}\right]dV.
$$
Independent thermal and <volume (thermodynamics)> variations give $T=T_R$ and
$$
\boxed{p_{\mathrm{air}}-p_R=\frac23\Gamma\alpha V_0^{-1/3},\qquad
V_0=\left[\frac{2\Gamma\alpha}{3(p_{\mathrm{air}}-p_R)}\right]^3.}
$$
The air <pressure> is evaluated at $V_0$; since the bubble is closed it generally depends on <volume (thermodynamics)>, so this equation is implicit until an air <equation of state> is supplied. For a <sphere>, $\alpha=(36\pi)^{1/3}$, $\Omega=4\pi R^2$, and the <pressure> jump becomes $2\Gamma/R$, the <Young–Laplace equation> for a single gas–liquid interface. There is no second surface as in a thin soap film.
\b[Isothermal <volume (thermodynamics)> fluctuations.] With the air maintained at $T_R$, eliminate its <entropy> in favor of its <Helmholtz free energy>:
$$
A_T(V)=F_{\mathrm{air}}(T_R,V,N)+p_RV+\Gamma\alpha V^{2/3}.
$$
The identity $(\partial F_{\mathrm{air}}/\partial V)_{T,N}=-p_{\mathrm{air}}$ gives
$$
A_T''(V_0)
=-\left(\frac{\partial p_{\mathrm{air}}}{\partial V}\right)_{T_R,N,V_0}
-\frac29\Gamma\alpha V_0^{-4/3}
=\frac1{V_0}\left[\kappa_T^{-1}-\frac29\Gamma\alpha V_0^{-1/3}\right].
$$
Using the curvature result,
$$
\boxed{\left\langle\Delta V^2\right\rangle\simeq
\frac{k_BT_RV_0}{\kappa_T^{-1}-\frac29\Gamma\alpha V_0^{-1/3}}.}
$$
These are the <volume fluctuations of a closed gas bubble>. Stability requires the denominator to be positive. The negative surface contribution reflects the concavity of $V^{2/3}$; gas compression supplies the positive restoring curvature. For ideal air, $\kappa_T^{-1}=p_{\mathrm{air}}$, and <mechanical equilibrium> reduces the denominator to
$$
\boxed{D=p_R+\frac49\Gamma\alpha V_0^{-1/3}>0.}
$$
The units are consistent: $D$ is <pressure>, so $k_BT_RV_0/D$ has units $\mathrm{m^6}$, the units of a <volume (thermodynamics)> <variance>. Holding $p_{\mathrm{air}}$ externally fixed while varying $V$ would remove the gas-compression term and change this stability problem; that is not the fixed-particle-number situation here.
\b[Numerical size of the fluctuations.] The <volume (thermodynamics)> conversion is important:
$$
V_0=10^{-3}\,\mathrm{mm^3}=10^{-12}\,\mathrm{m^3},\qquad
V_0^{1/3}=10^{-4}\,\mathrm m,\qquad T_R=298.15\,\mathrm K.
$$
Use the specified $\alpha=4.84$, $\Gamma=0.072\,\mathrm{N\,m^{-1}}$ and $p_R=10^5\,\mathrm{Pa}$. Then
$$
p_{\mathrm{air}}-p_R
=\frac23(0.072)(4.84)(10^4)\,\mathrm{Pa}
=2323.2\,\mathrm{Pa},
$$
$$
D=10^5+\frac49(0.072)(4.84)(10^4)
=101548.8\,\mathrm{Pa}.
$$
With the <Boltzmann constant> $k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$,
$$
k_BT_R=4.1164\times10^{-21}\,\mathrm J,
\qquad
\left\langle\Delta V^2\right\rangle\simeq4.05\times10^{-38}\,\mathrm{m^6}.
$$
Thus the fractional <root mean square> fluctuation and fractional <variance> are respectively
$$
\boxed{\frac{\sqrt{\langle\Delta V^2\rangle}}{V_0}\simeq2.01\times10^{-7},
\qquad\frac{\langle\Delta V^2\rangle}{V_0^2}\simeq4.05\times10^{-14}.}
$$
The absolute <root mean square> fluctuation is $2.01\times10^{-19}\,\mathrm{m^3}$. The very small fractional value supports the local <Gaussian fluctuation approximation>. This estimate neglects exchange of dissolved air, variations of <surface tension>, shape fluctuations and <temperature> fluctuations, consistently with the prescribed fixed-number, isothermal bubble model.
Back to article page
