Match the rate expressions in LIST-I for the decomposition of \(\mathrm{X}\) with the corresponding profiles provided in LIST-II. \(X_s\) and \(\mathrm{k}\) are constants having appropriate units.

$\left(\mathrm{I}\right)$ rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{S}}+\left[\mathrm{X}\right]}$

Case-1: $\left[\mathrm{X}\right]>>{\mathrm{X}}_{\mathrm{s}};\left[\mathrm{X}\right]+{\mathrm{X}}_{\mathrm{s}}\approx \left[\mathrm{X}\right]$

rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{\left[\mathrm{X}\right]}=\mathrm{k}$    (Zero order w.r.t. $\mathrm{X}$)

$\mathrm{I}\to \mathrm{P},\mathrm{S}$

 

Case-2: $\left[\mathrm{X}\right]\ll {\mathrm{X}}_{\mathrm{s}};\left[\mathrm{X}\right]+{\mathrm{X}}_{\mathrm{s}}\approx {\mathrm{X}}_{\mathrm{s}}$

$\therefore$ rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{S}}}={\mathrm{k}}^{\text{'}}\left[\mathrm{X}\right]$   ($1^{\mathrm{st}}$ order w.r.t. $\mathrm{X}$)

$\therefore \mathrm{I}\to \mathrm{Q},\mathrm{T}$

Case-3: $\left[\mathrm{X}\right]\approx {\mathrm{X}}_{\mathrm{S}}$

rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{s}}+\left[\mathrm{X}\right]}$

In this case curve-$\mathrm{R}$ given in List-II will match.

$\therefore \mathrm{I}\to \mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S},\mathrm{T}$  (The graph of half-life should start from origin)

$\left(\mathrm{II}\right)$ rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{S}}+\left[\mathrm{X}\right]}$

$\because \left[\mathrm{X}\right]\ll {\mathrm{X}}_{\mathrm{S}}$

$\therefore {\mathrm{X}}_{\mathrm{s}}+\left[\mathrm{X}\right]\approx {\mathrm{X}}_{\mathrm{s}}$

$\therefore$  rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{S}}}={\mathrm{k}}^{\text{'}}\left[\mathrm{X}\right]$  ($1^{\mathrm{st}}$ order w.r.t. $\mathrm{X}$)

$\therefore \mathrm{II}\to \mathrm{Q},\mathrm{T}$

$\left(\mathrm{III}\right)$  rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{{\mathrm{X}}_{\mathrm{s}}+\left[\mathrm{X}\right]}$

$\because \left[\mathrm{X}\right]>>{\mathrm{X}}_{\mathrm{s}}$

$\therefore {\mathrm{X}}_{\mathrm{s}}+\left[\mathrm{X}\right]\approx \left[\mathrm{X}\right]$

$\therefore$  rate $=\frac{\mathrm{k}\left[\mathrm{X}\right]}{\left[\mathrm{X}\right]}=\mathrm{k}$   (Zero order w.r.t. $\mathrm{X}$)

$\therefore \mathrm{III}\to \mathrm{P},\mathrm{S}$

$\left(\mathrm{IV}\right)$ rate $=\frac{\mathrm{k}{\left[\mathrm{X}\right]}^2}{{\mathrm{X}}_{\mathrm{s}}+\left[\mathrm{X}\right]}$

$\because \left[\mathrm{X}\right]>>{\mathrm{X}}_{\mathrm{S}}$

$\therefore {\mathrm{X}}_{\mathrm{S}}+\left[\mathrm{X}\right]\approx \left[\mathrm{X}\right]$

$\therefore$  rate $=\frac{\mathrm{k}{\left[\mathrm{X}\right]}^2}{\left[\mathrm{X}\right]}=\mathrm{k}\left[\mathrm{X}\right]$   ($1^{\mathrm{st}}$ order w.r.t. $\mathrm{X}$)

$\therefore \mathrm{IV}\to \mathrm{Q},\mathrm{T}$