up:: [[Chemical Kinetics]] Atoms/molecules do not react unless they collide. (There are some exceptions to this rule e.g. molecules break down on heating. But lets ignore that for this treatment) For a general reaction $\sum_{i=1}^{N}\ce{\nu_i^'m_i<=>}\sum_{i=1}^{N}\ce{\nu_i^{''}m_i} \tag{1} $ ## 1. Degree of advancement of reaction $\frac{dN_i}{\nu_i^{''}-\nu_i^{'}} = d\xi = constant$ $dN$ is the change in the number of moles The constant $d\xi$ is known as degree of advancement of reaction For example for $\ce{H_2 +O_2<=>H_2O}$ $\frac{dN_{H_2}}{-1}=\frac{dN_{O_2}}{-\frac{1}{2}}=\frac{dN_{H_2O}}{1}$ [[Reaction Rate]] ## 2. Rate of production/depletion of a species Rate of production/depletion of a species $\frac{dC_i}{dt}=\omega_i$ hence $\omega_i=\omega(\nu_i^{''}-\nu_i^{'})\tag{2}$ ## 3. Law of mass action Law of mass action can be written as $\omega=k\prod_{i=1}^N C_i^{(\nu_i^{''}-\nu_i^{'})}$ Note that in the above expression the reactants will come to the denominator and products will be in numerator ($\nu^{''}$ is lower than $\nu^{'}$ for reactants) For example, for a reaction $\ce{aA + bB<=> cC + dD}$ the law of mass action can be written as $\omega=k\frac{C_C^c C_D^d}{C_A^aC_B^b}$ $k$ is called *rate constant* but its not really a constant but rather a function of temperature. ## 4. Arrhenius Equation Arrhenius equation gives the temperature dependence of $k$ $k=Ae^{-E/R_uT}$ Where, A - Arrhenius constant - its also not a constant but a function of temperature $A=BT^m$ E - activation energy $R_u$ - universal gas constant T - absolute temperature $k=BT^me^{-E/R_uT}$ ## 5. Law of Mass action for a system of equations For a system of reactions containing M reactions $\sum_{i=1}^{N}\ce{\nu_{ik}^'m_i<=>}\sum_{i=1}^{N}\ce{\nu_{ik}^{''}m_i}\ k = 1\ to\ M, \tag{3} $ law of mass action for the $k^{th}$ reaction can be written as $\omega_k=k_k\prod_{i=1}^N C_i^{(\nu_{ik}^{''}-\nu_{ik}^{'})}$ $\omega_k=A_ke^{-E/R_uT}\prod_{i=1}^N C_i^{(\nu_{ik}^{''}-\nu_{ik}^{'})}$ for a given species $i$ $\omega_i=\sum\omega_{ik}$ Similar to equation (2) $\omega_{ik} = \omega_k(\nu_{ik}^{''}-\nu_{ik}^{'})$ Therefore, combining above reactions $\omega_i=\frac{dC_i}{dt}=\sum (\nu_{ik}^{''}-\nu_{ik}^{'})A_{ki}e^{-E/R_uT}\prod_{i=1}^N C_i^{(\nu_{ik}^{''}-\nu_{ik}^{'})}$ The intuition from the above equation is that the rate of change of a concentration of one species depends on the rate of change of concentration of all species in all the reactions in the system. It is a simultaneous set of first order non-linear differential equations. ## 6. Law of Mass action in equilibrium for a system of reactions At equilibrium $\omega_k = 0$ i.e forward reactions and backward reactions are happening at the same time $k_{fk}\prod_{i=1}^N C_i^{\nu_{ik}^{'}}=k_{bk}\prod_{i=1}^N C_i^{\nu_{ik}^{''}}$ or $\frac{k_{fk}}{k_{bk}}=\frac{\prod_{i=1}^N C_i^{\nu_{ik}^{''}}}{\prod_{i=1}^N C_i^{\nu_{ik}^{'}}}= k_{ck}$ $k_{ck}$ is the [Equilibrium constant](!https://byjus.com/jee/equilibrium-constant/#:~:text=For%20example%2C%20the%20equilibrium%20constant,to%20their%20respective%20stoichiometric%20coefficients.) based on concentration From ideal gas law applied to one species $i$, $P_iV=n_iRT$ or $\frac{n_i}{V}=\frac{P_i}{RT}=C_i$ From Dalton's law of partial pressures $P_i=x_iP$