Problem 79
Question
A substance reacts according to first-order kinetics. The rate constant for the reaction is \(1 \times\) \(10^{-2} \mathrm{sec}^{1}\). Its initial concentration is IM. Its initial rate is (a) \(2 \times 10^{2} \mathrm{Ms}^{-1}\) (b) \(1 \times 10^{2} \mathrm{Ms}^{-1}\) (c) \(1 \times 10^{-2} \mathrm{Ms}^{-1}\) (d) \(2 \times 10^{-2} \mathrm{Ms}^{-1}\)
Step-by-Step Solution
Verified Answer
(c) \(1 \times 10^{-2} \mathrm{Ms}^{-1}\).
1Step 1: Understand First-Order Reaction
In a first-order reaction, the rate of reaction depends linearly on the concentration of one reactant. The general expression for the rate of a first-order reaction is given by \( \text{Rate} = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.
2Step 2: Substitute Known Values
The rate constant \( k \) is given as \( 1 \times 10^{-2} \, \text{sec}^{-1} \) and the initial concentration \( [A] = 1 \, \text{M} \). Substitute these values into the first-order rate equation: \[ \text{Rate} = (1 \times 10^{-2}) \times (1) \]
3Step 3: Calculate Initial Rate of Reaction
Solve the equation to find the initial rate:\[ \text{Rate} = 1 \times 10^{-2} \, \text{M sec}^{-1} \]
4Step 4: Choose the Correct Option
Compare the calculated initial rate \( 1 \times 10^{-2} \, \text{M sec}^{-1} \) with the options provided. The correct answer is (c) \( 1 \times 10^{-2} \, \text{M sec}^{-1} \).
Key Concepts
Understanding Rate ConstantRole of Concentration in KineticsUnderstanding Rate of Reaction
Understanding Rate Constant
The rate constant, often denoted by the symbol \( k \), is a crucial component in the realm of chemical kinetics. It serves as a proportionality factor in the rate equation for a reaction. For first-order reactions, the rate constant has a unique dependency. Here, the reaction rate depends directly on the concentration of a single reactant.
The rate constant has specific units depending on the order of the reaction. For a first-order reaction, these units are typically expressed as \( \text{sec}^{-1} \). This signifies how the reaction rate is related to time rather than concentration.
In our exercise, the rate constant is given as \( 1 \times 10^{-2} \text{ sec}^{-1} \). This number helps us understand the speed at which the reaction proceeds, independent of concentration. Key takeaway? A higher \( k \) value signifies a faster reaction under the same conditions.
The rate constant has specific units depending on the order of the reaction. For a first-order reaction, these units are typically expressed as \( \text{sec}^{-1} \). This signifies how the reaction rate is related to time rather than concentration.
In our exercise, the rate constant is given as \( 1 \times 10^{-2} \text{ sec}^{-1} \). This number helps us understand the speed at which the reaction proceeds, independent of concentration. Key takeaway? A higher \( k \) value signifies a faster reaction under the same conditions.
Role of Concentration in Kinetics
Concentration refers to the amount of a substance in a certain volume of solution, typically expressed as molarity (M), which is moles per liter.
In the context of first-order reactions, the concentration of a reactant plays a direct role in determining the rate of reaction. The expression for the rate of a first-order reaction is \( \text{Rate} = k[A] \). Here, \([A]\) is the concentration of reactant A, which directly impacts how fast the reaction occurs.
When the concentration changes, the rate of reaction changes proportionally. For instance, if you were to double the concentration of the reactant, the rate of the reaction would also double, assuming that the rate constant remains constant.
In our problem, the initial concentration of reactant A is 1 M, showing us that the starting concentration has a tangible effect on how we calculate the initial rate. Remember, first-order reactions are particularly dependent on the concentration remaining steady during the reaction!
In the context of first-order reactions, the concentration of a reactant plays a direct role in determining the rate of reaction. The expression for the rate of a first-order reaction is \( \text{Rate} = k[A] \). Here, \([A]\) is the concentration of reactant A, which directly impacts how fast the reaction occurs.
When the concentration changes, the rate of reaction changes proportionally. For instance, if you were to double the concentration of the reactant, the rate of the reaction would also double, assuming that the rate constant remains constant.
In our problem, the initial concentration of reactant A is 1 M, showing us that the starting concentration has a tangible effect on how we calculate the initial rate. Remember, first-order reactions are particularly dependent on the concentration remaining steady during the reaction!
Understanding Rate of Reaction
The rate of reaction is a measurement of the change in concentration of reactants or products over time. In the context of first-order reactions, this rate can be easily calculated using the formula \( \text{Rate} = k[A] \).
Here, the rate is expressed in terms of \( \text{M}\,\text{sec}^{-1} \), highlighting how the concentration changes every second. Initially, the rate of reaction provides a snapshot of the reaction pace when it begins. Calculating this rate can tell us a lot about how quickly a reactant is being consumed or a product is being formed.
In our specific scenario, the initial rate computed was \( 1 \times 10^{-2} \text{ M sec}^{-1} \). This number helps put the reaction speed into perspective—how rapidly the concentration of the reactant is diminishing at the very start.
Here, the rate is expressed in terms of \( \text{M}\,\text{sec}^{-1} \), highlighting how the concentration changes every second. Initially, the rate of reaction provides a snapshot of the reaction pace when it begins. Calculating this rate can tell us a lot about how quickly a reactant is being consumed or a product is being formed.
In our specific scenario, the initial rate computed was \( 1 \times 10^{-2} \text{ M sec}^{-1} \). This number helps put the reaction speed into perspective—how rapidly the concentration of the reactant is diminishing at the very start.
- Why Calculate Initial Rates? The initial rate gives insight into the reaction dynamics before any significant changes in concentration occur.
- Changes over Time: As the reaction proceeds, the concentration of the reactant typically decreases, which in turn affects the rate.
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