Cheat sheet

Every key formula, in canonical order, at a glance. Each formula links to the lesson that derives it.

1 — The liquid state

U(r) = 4\varepsilon\left[(\sigma_0/r)^{12} - (\sigma_0/r)^{6}\right]

r_\text{eq} = 2^{1/6}\sigma_0

K = -V\left.\dfrac{\partial p}{\partial V}\right\rvert_T

\dfrac{\Delta V}{V} = -\dfrac{\Delta p}{K}

p_\text{coh} \approx -0.235\,K

\left.\dfrac{\partial p}{\partial V}\right\rvert_T = 0

W \approx \dfrac{p^2 V}{2K}

\Delta G(R) = -\tfrac{4}{3}\pi R^3(p_v - p_\infty) + 4\pi R^2\sigma

R^* = \dfrac{2\sigma}{\Delta p}

\Delta G^* = \dfrac{16\pi\sigma^3}{3(\Delta p)^2}

J = J_0\exp\!\left(-\dfrac{\Delta G^*}{k_B T}\right)

p_g - p_\infty = \dfrac{2\sigma}{R_m}

\theta + \beta > 90^\circ

p_{\infty,\text{crit}} = p_g - \dfrac{2\sigma}{r_c}\cos(\theta+\beta-90^\circ)

N(R) \sim R^{-4}

p_g + p_v - p_\infty = \dfrac{2\sigma}{R_0}

f(p_\infty) = \displaystyle\int_{R_\text{crit}(p_\infty)}^{\infty} N(R)\,dR

c_g = k_H\,p_{g,\text{gas-phase}}

\sigma_i = \dfrac{p_\infty^* - p_v}{\tfrac{1}{2}\rho U_\infty^2} = -C_{p,\text{min}}

u(r,t) = \dfrac{R^2\dot R}{r^2}

\rho\left[R\ddot R + \tfrac{3}{2}\dot R^2\right] = p_B - p_\infty - \dfrac{2\sigma}{R} - \dfrac{4\mu\dot R}{R}

p_B - p_\text{liq,wall} = \dfrac{2\sigma}{R}

p_B(t) = p_v + p_{G,0}\left(\dfrac{R_0}{R}\right)^{3\kappa}

\dfrac{dp_v}{dT} = \dfrac{L\rho_v}{T}

\tau_\text{th} \approx \dfrac{R^2}{\alpha_\text{gas}}

p_\infty = p_v + \dfrac{K}{R^3} - \dfrac{2\sigma}{R}

R_B = \sqrt{\dfrac{3K}{2\sigma}}

p_{\infty,\text{crit}} = p_v - \dfrac{4}{3}\sqrt{\dfrac{2\sigma^3}{3K}}

\omega_0 = \dfrac{1}{R_0}\sqrt{\dfrac{3\kappa p_{G,0} - 2\sigma/R_0}{\rho}}

\dot R^2 = \dfrac{2p_\infty}{3\rho}\left[\left(\dfrac{R_{\max}}{R}\right)^3 - 1\right]

\tau_\text{Rayleigh} \approx 0.915\,R_{\max}\sqrt{\dfrac{\rho}{p_\infty}}

\dot R_\infty = \sqrt{\dfrac{2\lvert p_\infty\rvert}{3\rho}}