Explain the integrated rate equation for a first-order reaction. Derive the expression for the rate constant of a first-order reaction and also discuss the characteristics of first-order reactions along with the concept of half-life period.

Q: Explain the integrated rate equation for a first-order reaction. Derive the expression for the rate constant of a first-order reaction and also discuss the characteristics of first-order reactions along with the concept of half-life period.

Definition and Integrated Rate Equation\nA first-order reaction is one in which the rate of the reaction is directly proportional to the first power of the concentration of the reactant. Consider a general first-order reaction: $R \rightarrow P$. \nThe differential rate law is given by: $$\text{Rate} = -\frac{dR}{dt} = kR$$ \nRearranging the equation to separate the variables: $$\frac{dR}{R} = -k \cdot dt$$ \nIntegrating both sides of the equation between the limits $t = 0$ (initial time where concentration is $R0$) and time $t$ (where concentration is $Rt$): $$\int{R0}^{Rt} \frac{dR}{R} = -k \int{0}^{t} dt$$ $$\lnRt - \lnR0 = -kt$$ $$\ln\frac{Rt}{R0} = -kt$$ or $$k = \frac{1}{t} \ln\frac{R0}{Rt}$$ \nConverting natural logarithm to base 10: $$k = \frac{2.303}{t} \log\frac{R0}{Rt}$| Characteristics of First-Order Reactions Units of Rate Constant: The unit of rate constant ($k$) for a first-order reaction is $\text{time}^{-1}$ (e.g., $\text{s}^{-1}$, $\text{min}^{-1}$). Concentration Dependence: The rate of the reaction depends directly on the concentration of the reactant. If concentration is doubled, the rate also doubles. Linear Graph: A plot of $\logRt$ versus time $t$ gives a straight line with a slope equal to $-\frac{k}{2.303}$ and an intercept equal to $\logR0$. Half-Life Period ($t{1/2}$)\nThe half-life of a reaction is the time in which the concentration of the reactant is reduced to one half of its initial concentration. At $t = t{1/2}$, $Rt = \frac{R0}{2}$. Substituting these values into the integrated rate equation: $$k = \frac{2.303}{t{1/2}} \log\frac{R0}{R0 / 2}$$ $$k = \frac{2.303}{t{1/2}} \log 2$$ \nSince $\log 2 = 0.3010$: $$k = \frac{2.303 \times 0.3010}{t{1/2}} = \frac{0.693}{t{1/2}}$$ \nTherefore, the half-life of a first-order reaction is independent of the initial concentration of the reactant.

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Definition and Integrated Rate Equation
A first-order reaction is one in which the rate of the reaction is directly proportional to the first power of the concentration of the reactant. Consider a general first-order reaction: $R ightarrow P$. 
The differential rate law is given by: $$	ext{Rate} = -\frac{dR}{dt} = kR$$ 
Rearranging the equation to separate the variables: $$\frac{dR}{R} = -k \cdot dt$$ 
Integrating both sides of the equation between the limits $t = 0$ (initial time where concentration is $R0$) and time $t$ (where concentration is $Rt$): $$\int{R0}^{Rt} \frac{dR}{R} = -k \int{0}^{t} dt$$ $$\lnRt - \lnR0 = -kt$$ $$\ln\frac{Rt}{R0} = -kt$$ or $$k = \frac{1}{t} \ln\frac{R0}{Rt}$$ 
Converting natural logarithm to base 10: $$k = \frac{2.303}{t} \log\frac{R0}{Rt}$| Characteristics of First-Order Reactions Units of Rate Constant: The unit of rate constant ($k$) for a first-order reaction is $	ext{time}^{-1}$ (e.g., $	ext{s}^{-1}$, $	ext{min}^{-1}$). Concentration Dependence: The rate of the reaction depends directly on the concentration of the reactant. If concentration is doubled, the rate also doubles. Linear Graph: A plot of $\logRt$ versus time $t$ gives a straight line with a slope equal to $-\frac{k}{2.303}$ and an intercept equal to $\logR0$. Half-Life Period ($t{1/2}$)
The half-life of a reaction is the time in which the concentration of the reactant is reduced to one half of its initial concentration. At $t = t{1/2}$, $Rt = \frac{R0}{2}$. Substituting these values into the integrated rate equation: $$k = \frac{2.303}{t{1/2}} \log\frac{R0}{R0 / 2}$$ $$k = \frac{2.303}{t{1/2}} \log 2$$ 
Since $\log 2 = 0.3010$: $$k = \frac{2.303 	imes 0.3010}{t{1/2}} = \frac{0.693}{t{1/2}}$$ 
Therefore, the half-life of a first-order reaction is independent of the initial concentration of the reactant.

MP Board · Class 12 · Chemistry · Chemical KineticsExplain the integrated rate equation for a first-order reaction. Derive the expression for the rate constant of a first-order reaction and also discuss the characteristics of first-order reactions along with the concept of half-life period.

Definition and Integrated Rate Equation\nA first-order reaction is one in which the rate of the reaction is directly proportional to the first power of the concentration of the reactant. Consider a general first-order reaction: $R \rightarrow P$.

Characteristics of First-Order Reactions

Half-Life Period ($t_{1/2}$)\nThe half-life of a reaction is the time in which the concentration of the reactant is reduced to one half of its initial concentration. At $t = t_{1/2}$, $[R]_t = \frac{[R]_0}{2}$. Substituting these values into the integrated rate equation: