Zero and First Order Reactions — Revision Notes

⚡ 30-Second Revision

Zero-Order Reaction:

- Rate Law: $\text{Rate} = k$ - Integrated Rate Law: $[A]_t = [A]_0 - kt$ - Half-life: $t_{1/2} = \frac{[A]_0}{2k}$ (proportional to $[A]_0$) - Units of $k$: mol L$^{-1}$ s$^{-1}$ - Linear Plot: $[A]_t$ vs. $t$ (slope = $-k$)

First-Order Reaction:

- Rate Law: $\text{Rate} = k[A]$ - Integrated Rate Law: $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$ or $2.303 \log\left(\frac{[A]_0}{[A]_t}\right) = kt$ - Half-life: $t_{1/2} = \frac{0.693}{k}$ (independent of $[A]_0$) - Units of $k$: s$^{-1}$ - Linear Plot: $\ln[A]_t$ vs. $t$ (slope = $-k$)

2-Minute Revision

For NEET, quickly recall the core aspects of zero and first-order reactions. Zero-order reactions have a constant rate, meaning the reactant's concentration doesn't affect how fast it reacts. Think of it as a factory producing at a fixed pace, regardless of how much raw material is stockpiled.

The key formula is $[A]_t = [A]_0 - kt$. Its half-life, $t_{1/2} = [A]_0 / 2k$, is directly proportional to the initial concentration. The rate constant ($k$) for zero-order reactions has units of mol L$^{-1}$ s$^{-1}$.

Graphically, a plot of concentration vs. time is a straight line with a negative slope of $-k$.

First-order reactions, on the other hand, have a rate directly proportional to the reactant's concentration. Imagine a population growth where the more individuals there are, the faster it grows. The integrated rate law is $ln([A]_t/[A]_0) = -kt$.

A crucial feature is its half-life, $t_{1/2} = 0.693 / k$, which is constant and independent of the initial concentration. The units of $k$ for first-order reactions are s$^{-1}$. A plot of $ln[A]_t$ vs.

time yields a straight line with a negative slope of $-k$. Remember these distinct characteristics, especially the half-life dependence and rate constant units, as they are frequently tested.

5-Minute Revision

A thorough revision of zero and first-order reactions for NEET involves understanding their definitions, integrated rate laws, half-life characteristics, and graphical representations.

Zero-Order Reactions: The rate is independent of reactant concentration. This implies that the reaction proceeds at a constant speed until the reactant is depleted. The differential rate law is $\text{Rate} = -d[A]/dt = k$.

Integrating this gives the integrated rate law: $[A]_t = [A]_0 - kt$. This equation is linear, so a plot of $[A]_t$ vs. $t$ yields a straight line with a slope of $-k$ and y-intercept $[A]_0$. The half-life for a zero-order reaction is $t_{1/2} = [A]_0 / 2k$, which means it is directly proportional to the initial concentration.

The **units of the rate constant ($k$)** are mol L$^{-1}$ s$^{-1}$. A common example is the decomposition of ammonia on a hot platinum surface when the surface is saturated.

First-Order Reactions: The rate is directly proportional to the first power of the reactant concentration. This means the reaction slows down as the reactant is consumed. The differential rate law is $\text{Rate} = -d[A]/dt = k[A]$.

Integrating this gives the integrated rate law: $ln([A]_t/[A]_0) = -kt$ or, in base-10 logarithm, $2.303 \log([A]_0/[A]_t) = kt$. A plot of $ln[A]_t$ vs. $t$ (or $log[A]_t$ vs. $t$) yields a straight line with a slope of $-k$ (or $-k/2.

303$) and y-intercept$ln[A]_0$(or$log[A]_0$). The most distinctive feature is its **half-life**:$t_{1/2} = 0.693 / k$, which is constant and independent of the initial concentration. The **units of the rate constant ($k$)** are s$^{-1}$.

Radioactive decay is a classic example.

Key Differences to Remember: Focus on the dependence of half-life on initial concentration and the units of $k$. Practice numerical problems involving both types, especially those requiring calculation of time, concentration, or rate constant. Be prepared to interpret graphs to determine reaction order.

Prelims Revision Notes

Zero-Order Reactions

Definition: — Rate is independent of reactant concentration.
Rate Law: — $\text{Rate} = k[A]^0 = k$
Integrated Rate Law: — $[A]_t = [A]_0 - kt$

* $[A]_t$: concentration at time $t$ * $[A]_0$: initial concentration * $k$: rate constant * $t$: time

Units of Rate Constant ($k$): — mol L$^{-1}$ s$^{-1}$ (or M s$^{-1}$)
Half-life ($t_{1/2}$): — $t_{1/2} = \frac{[A]_0}{2k}$

* Key Feature: $t_{1/2}$ is directly proportional to the initial concentration $[A]_0$.

Graphical Representation: — Plot of $[A]_t$ vs. $t$ is a straight line.

* Slope $= -k$ * Y-intercept $= [A]_0$

Examples: — Enzyme-catalyzed reactions (at high substrate conc.), surface-catalyzed reactions (e.g., $NH_3$ decomposition on Pt).

First-Order Reactions

Definition: — Rate is directly proportional to the first power of reactant concentration.
Rate Law: — $\text{Rate} = k[A]^1 = k[A]$
Integrated Rate Law:

* $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$ * $2.303 \log\left(\frac{[A]_0}{[A]_t}\right) = kt$ * $[A]_t = [A]_0 e^{-kt}$

Units of Rate Constant ($k$): — s$^{-1}$ (or min$^{-1}$, hr$^{-1}$)
Half-life ($t_{1/2}$): — $t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{k}$

* Key Feature: $t_{1/2}$ is constant and independent of the initial concentration $[A]_0$.

Graphical Representation: — Plot of $ln[A]_t$ vs. $t$ is a straight line.

* Slope $= -k$ * Y-intercept $= \ln[A]_0$ * Plot of $log[A]_t$ vs. $t$ is a straight line with slope $= -k/2.303$.

Examples: — Radioactive decay, decomposition of $N_2O_5$, hydrolysis of esters (pseudo-first-order).

Important Points for NEET

Distinguish Order vs. Molecularity: — Order is experimental, molecularity is theoretical (for elementary steps).
Pseudo-First-Order Reactions: — Understand when a higher-order reaction behaves as first-order (e.g., excess solvent).
Calculations: — Be proficient in using integrated rate laws and half-life formulas to solve for $k$, $t$, $[A]_t$, or $[A]_0$. Remember $ln(x) = 2.303 \log(x)$.

Vyyuha Quick Recall

Zero Constant Linear Half-life Proportional

Zero-order: Constant rate (independent of concentration)
Linear plot: $[A]$ vs $t$
Half-life: Proportional to initial concentration ($[A]_0$)

First Exponential Log Half-life Independent

First-order: Exponential decay
Log plot: $\ln[A]$ vs $t$
Half-life: Independent of initial concentration