write an equation relating the concentration of a reactant A at t=0 to that of t=t for a first order reaction. Define all the terms and give their units. Do the same for a second order reaction.
To derive the equation relating the concentration of a reactant A at t=0 to that at t=t for a first-order reaction, we will use the integrated rate law for first-order reactions, which is given by the equation:
ln(A[t]/A[0]) = -kt
Here:
- A[t] represents the concentration of reactant A at time t.
- A[0] represents the initial concentration of reactant A at t=0.
- k is the rate constant of the first-order reaction.
- t is the time elapsed since the start of the reaction.
To find the concentration of reactant A at t=0, we substitute t=0 into the equation:
ln(A[0]/A[0]) = -k(0)
ln(1) = 0
Therefore, the equation simplifies to:
0 = 0
This indicates that the concentration of reactant A at t=0 is equal to its initial concentration, A[0].
Now, let's move on to the second-order reaction.
The integrated rate law for a second-order reaction is given by the equation:
1/A[t] - 1/A[0] = kt
Here:
- A[t] represents the concentration of reactant A at time t.
- A[0] represents the initial concentration of reactant A at t=0.
- k is the rate constant of the second-order reaction.
- t is the time elapsed since the start of the reaction.
Similar to the first-order reaction, we want to find the concentration of reactant A at t=0 by substituting t=0 into the equation:
1/A[0] - 1/A[0] = k(0)
0 = 0
This shows that the concentration of reactant A at t=0 is also equal to its initial concentration, A[0].
For the units:
- The concentration of reactant A is usually measured in units such as moles per liter (mol/L) or molarity (M).
- The time, t, is typically measured in seconds (s), minutes (min), or any other time unit as long as it is consistent.
- The rate constant, k, will have units that depend on the overall order of the reaction.
- For a first-order reaction, the units of k will be s^(-1) or min^(-1), depending on the time units used.
- For a second-order reaction, the units of k will be L/(mol·s) or L/(mol·min), again depending on the time units used.