These things directly follow from the rate law. We can derive the integrated rate laws by integrating those equations. I’m going to show these derivations in case you’re curious and(or) you’ve taken calculus and(or) why not.
Zero-order reactions
r=−dtd[A]−dtd[A]−d[A]−∫[A]t1[A]t2d[A]−[A][A]t1[A]t2−([A]t2−[A]t1)=k[A]0=k=kdt=∫t1t2kdt=ktt1t2=k(t2−t1)anything raised to 0 is 1moving things
Let’s call t1=0 and t2=t.
−([A]t−[A]0)[A]t−[A]0=kt=−kt
As we have on slide 21.
First-order reactions
r=−dtd[A]−[A]d[A]−∫[A]t1[A]t2[A]d[A]−ln[A][A]t1[A]t2−(ln[A]t2−ln[A]t1)ln[A]t1−ln[A]t2ln[A]t2[A]t1ln[A]t[A]0=k[A]=kdt=∫t1t2kdt=ktt1t2=k(t2−t1)=k(t2−t1)=k(t2−t1)=ktmoving thingsrenaming t1=0 and t2=t
Note that we can exponentiate both sides of the last line to get:
[A]t[A]0=ekt⟹[A]t=[A]0e−kt
from which follows that the concentration in the first-order reactions decays exponentially. That’s why the first-order reactions are often also called exponential decays.
Second-order reactions
r=−dtd[A]−[A]2d[A]−∫[A]t1[A]t2[A]2d[A][A]1[A]t1[A]t2[A]t21−[A]t11[A]t1−[A]01=k[A]2=kdt=∫t1t2kdt=ktt1t2=k(t2−t1)=ktmoving thingsrenaming t1=0 and t2=t