A first-order reaction has a half-life of 693 seconds. What will be its rate constant?
0.001 sec⁻¹
Understanding First-Order Reaction Kinetics and Half-Life
Chemical kinetics is the study of reaction rates. For a chemical reaction, the rate constant (\(k\)) is a proportionality constant that relates the reaction rate to the concentrations of reactants. The order of a reaction describes how the rate depends on the concentration of reactants. A first-order reaction is one where the reaction rate is directly proportional to the concentration of one reactant.
Calculating Rate Constant from Half-Life for a First-Order Reaction
The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For a first-order reaction, the half-life is constant and independent of the initial concentration of the reactant. There is a specific relationship between the half-life (\(t_{1/2}\)) and the rate constant (\(k\)) for a first-order reaction. This relationship is given by the formula:
\(t_{1/2} = \frac{0.693}{k}\)
We are given that the first-order reaction has a half-life of 693 seconds. We need to find its rate constant (\(k\)). We can rearrange the formula to solve for \(k\):
\(k = \frac{0.693}{t_{1/2}}\)
Step-by-Step Calculation
Given:
- Half-life, \(t_{1/2} = 693\) seconds
We need to find the rate constant, \(k\).
Using the formula for a first-order reaction:
\(k = \frac{0.693}{t_{1/2}}\)
Substitute the given value of \(t_{1/2}\):
\(k = \frac{0.693}{693 \text{ seconds}}\)
Now, perform the calculation:
\(k = 0.001 \text{ sec}^{-1}\)
The unit for the rate constant of a first-order reaction is typically time\(\text{⁻¹}\) (e.g., s\(\text{⁻¹}\), min\(\text{⁻¹}\), hr\(\text{⁻¹}\)). In this case, since the half-life is in seconds, the rate constant will have units of seconds\(\text{⁻¹}\).
Comparing with Options
The calculated rate constant is 0.001 sec\(\text{⁻¹}\). Let's look at the options provided:
- 0.001 sec\(\text{⁻¹}\)
- 1 sec\(\text{⁻¹}\)
- 0.001 sec\(\text{⁻¹}\)
- 0.1 sec\(\text{⁻¹}\)
Our calculated value matches option 1 and option 3.
Revision Table: First-Order Reaction Key Formulas
| Concept | Formula | Notes |
|---|---|---|
| Rate Law | \(\text{Rate} = k[\text{A}]\) | For reaction A \(\rightarrow\) Products |
| Integrated Rate Law | \(\ln[\text{A}]_t = \ln[\text{A}]_0 - kt\) | \([\text{A}]_t\) is concentration at time t, \([\text{A}]_0\) is initial concentration |
| Integrated Rate Law (Alternative form) | \(\ln\left(\frac{[\text{A}]_0}{[\text{A}]_t}\right) = kt\) | |
| Half-Life (\(t_{1/2}\)) | \(t_{1/2} = \frac{0.693}{k}\) | Independent of initial concentration |
| Rate Constant (\(k\)) from Half-Life | \(k = \frac{0.693}{t_{1/2}}\) | Derived from the half-life formula |
Additional Information on Reaction Order and Half-Life
The relationship between half-life and rate constant depends on the order of the reaction. While the half-life of a first-order reaction is constant, this is not true for other reaction orders.
- Zero-Order Reaction: The half-life of a zero-order reaction is dependent on the initial concentration. The formula is \(t_{1/2} = \frac{[\text{A}]_0}{2k}\).
- Second-Order Reaction: The half-life of a second-order reaction is also dependent on the initial concentration. The formula is \(t_{1/2} = \frac{1}{k[\text{A}]_0}\).
This makes the first-order reaction's constant half-life a unique and important characteristic.
Important Questions from Chemical Kinetics
Arrange the following in increasing order of their osmotic pressure generation at 298 K:
(The cell wall is permeable to water and not to the solute molecules)
(A) If a cell containing 0.5 moles of solute dissolved in 1 L of water is immersed in pure water.
(B) If a cell containing 0.25 moles of solute dissolved in 1 L of water is immersed in pure water.
(C) If a cell containing 0.1 moles of solute dissolved in 0.01 L of water is immersed in pure water.
(D) If a cell containing 0.2 moles of solute dissolved in 0.05 L of water is immersed in pure water.
Choose the correct answer from the options given below:
Arrange the following rate constant units in increasing order of their order of reaction:
(A) sec-1
(B) mol L-1 sec-1
(C) mol-1 L sec-1
(D) mol-2 L2 sec-1
Choose the correct answer from the options given below:
Which factor in Arrhenius equation corresponds to the fraction of molecules having kinetic energy greater than activation energy?
A reaction takes 30 minutes to complete 50% of the reaction and takes 45 minutes to complete 75% of the reaction. The order of the reaction is:
Ferric oxide in blast furnace's upper half is mainly reduced by:
