User:ScottTParker/Sandbox

$$ X_{micelle} = \frac{X_N}{N}$$ $$X_1$$ $$ C = X_1 + X_N \ $$

$$ X_1=\left( \frac{C-X_1}{N} \right)^{1/N}e^{-\left(\mu_1^0-\mu_N^0 \right) /kT}$$

$$ X_N=N \left[X_1e^{\left(\mu_1^0-\mu_N^0 \right) /kT} \right] ^N$$

$$ C_{CMC}=\frac{2}{N^{1/(N-1)}K^{N/(N-1)}}$$

$$K=e^{\left(\mu_1^0-\mu_N^0 \right) /kT}$$

$$ X = \frac{1}{A_{particle}}$$

$$ P = \frac{k T}{N} \left[ C + (N-1) X_1 \right] $$

$$ \pi = \frac{k T}{N} \left[ \frac{1}{(A-A_0)} + \frac{(N-1)}{(A_1-A_0)} \right] $$

$$ (A_1-A_0) = (A - A_0) \left[ 1 + \left(\frac{2 (A_c-A_0)}{A_1-A_0} \right)^{N-1} \right] $$

$$ E = E_{int} + E_{disp} + P A\ $$

$$ \frac{Kc}{\Delta R(\theta, c)}=\frac{1}{M_wP(\theta)}+2A_2c = \frac{1}{M_w}\left(1+ \frac{q^2 R_g^2}{3} \right)+2A_2c$$