Winter $2024$ Term

Equation Sheet

${}_R=\frac{1}{N}\frac{dN}{dt}$

$q_p=\frac{1}{x}\frac{dP}{dt}=\frac{1}{x}\frac{P}{t}$

$q_s=-\frac{1}{x}\frac{dS}{dt}=-\frac{1}{x}\frac{S}{t}$

$\frac{dx}{dt}={}_{\text{n}\text{e}\text{t}}x,x=x_0,at,t=0,ln(\frac{x}{x_0})={}_{\text{n}\text{e}\text{t}}t,or,x=x_0{e}^{{}_{\text{n}\text{e}\text{t}}t}$

$Y_{\frac{x}{S}}-=-\frac{x}{S},{}_g=\frac{{}_{m}S}{K_S+S}$

$\frac{1}{Y_{\frac{x}{A}TP}^{AP}}=\frac{1}{Y_{\frac{x}{A}TP}^M}+\frac{m_{\text{A}\text{T}\text{P}}}{D}$

$x_{\text{m}\text{a}\text{x}}=x_0+Y{S}_0$

$\frac{1}{Y_{\frac{x}{O_2}}^{AP}}=\frac{1}{Y_{\frac{x}{O_2}}^M}+\frac{m_{O_2}}{D}$

moles of $C{O}_2$ produced$]/[$moles of $O_2$ consumed$]$

$N_i=\frac{m_i}{M_i},N_T=N_i=N_c+N_N+N_o+N_H+N_P+N_{},x_i=\frac{N_i}{N_T}$

$C_{xC}{H}_{xH}{N}_{xN}{O}_{xO}{P}_{xP}{S}_{xS}$

$G=G_i-b_j{G}_j$

$t_c=\frac{1}{{}_m}ln(\frac{x_m}{x_0})+t_l$

$r_b=\frac{Y_{\frac{x}{S}}{S}_0}{(\frac{1}{{}_m})ln(\frac{x_m}{x_0})+t_l},r_{c,\text{o}\text{p}\text{t}}={}_m{Y}_{\frac{x}{S}}{S}_0,\frac{r_{c,\text{o}\text{p}\text{t}}}{r_b}=ln(\frac{x_m}{x_0})+{}_m{t}_l$

${}_g-k_d={}_{\text{n}\text{e}\text{t}}=(1+-C)D=[1+(1-C)]D$

$x_1=\frac{D}{{}_g}{Y}_{\frac{x}{S}}^M(S_0-S),{}_n=\frac{1}{D_n}=\frac{S_n-S_{n-1}}{r_{s,n}}$

$Da=\frac{\text{m}\text{a}\text{x}\text{i}\text{m}\text{u}\text{m}\text{r}\text{a}\text{t}\text{e}of\text{b}\text{i}\text{o}\text{c}\text{o}\text{n}\text{v}\text{e}\text{r}\text{s}\text{i}\text{o}\text{n}}{\text{m}\text{a}\text{x}\text{i}\text{m}\text{u}\text{m}\text{r}\text{a}\text{t}\text{e}of\text{d}\text{i}\text{f}\text{f}\text{u}\text{s}\text{i}\text{o}\text{n}}=\frac{r_{\text{m}\text{a}\text{x}}}{(\frac{D_e}{})S_0}$

OUR$=x*q_{O_2}=k_{L}a(C^{**}-C_L),k_d=-\frac{d}{dt}lnp(t),k_d=e^{-\frac{E_{0d}}{R}T}$

$H(x_0-x_1)=L{Y}_1,\%$ extraction $=1-\frac{1}{R+1},{}_{ij}=\frac{K_i}{K_j}$