The interaction energy between an ion pair diminishes with separation distance r as a function of 1/r, whereas the interaction energy between two rotating dipoles diminishes as 1/r6. This means that ion-ion interactions are much longer range than dipole-dipole interactions. In fact, all so-called "van der Waals" interactions diminish as 1/r6, whether it be dipole-dipole, dipole-induced dipole, or induced dipole-induced dipole (London). To compare the range of interaction strength, compute the value of r when 1/r is only 1.0% of its value at r=1.
At that point, r =
Now compute the value of r when 1/r6 is only 1.0% of its value at r=1. Then r =
Those numbers are roughly how many Angstroms away from molecular contact that the force of interaction operates.
To compare the range of interaction strength, we need to compute the value of r when the expression 1/r is only 1.0% of its value at r=1.
To do this, we can set up the following equation:
1/r = (1.0%)*(1/r)|r=1
Simplifying, we have:
1/r = (0.01)*(1/1)
1/r = 0.01
To solve for r, we can take the reciprocal of both sides:
r = 1/0.01
r = 100
Therefore, when 1/r is only 1.0% of its value at r=1, r is equal to 100.
Now let's compute the value of r when the expression 1/r^6 is only 1.0% of its value at r=1.
We set up the following equation:
1/r^6 = (1.0%)*(1/r^6)|r=1
Simplifying, we have:
1/r^6 = (0.01)*(1/1^6)
1/r^6 = 0.01
To solve for r, we can take the sixth root of both sides:
r^6 = 1/0.01
r^6 = 100
Taking the sixth root of both sides, we get:
r = ∛(100)
r ≈ 4.6416
Therefore, when 1/r^6 is only 1.0% of its value at r=1, r is approximately 4.6416.
These values indicate that the ion-ion interaction operates at a much longer range than the dipole-dipole interaction. At 1.0% of its value at r=1, the ion-ion interaction operates at a distance of 100 angstroms away from molecular contact, while the dipole-dipole interaction operates at a distance of approximately 4.6416 angstroms away from molecular contact.
To compare the range of interaction strength, let's solve for the values of r when 1/r is 1.0% of its value at r=1, and when 1/r^6 is 1.0% of its value at r=1.
1. For the ion-ion interaction, the energy diminishes as 1/r. We want to find the value of r when 1/r is only 1.0% of its value at r=1.
1/r = 0.01(1)
To solve for r, we take the reciprocal of both sides:
r = 1/0.01
r = 100 Angstroms
Therefore, the value of r when 1/r is only 1.0% of its value at r=1 in the ion-ion interaction is approximately 100 Angstroms.
2. For dipole-dipole interactions, the energy diminishes as 1/r^6. We want to find the value of r when 1/r^6 is only 1.0% of its value at r=1.
1/r^6 = 0.01(1/1^6)
Simplifying, we have:
1/r^6 = 0.01
To solve for r, we take the sixth root of both sides:
r^6 = 1/0.01
r^6 = 100
r = ∛100
r ≈ 4.64 Angstroms
Therefore, the value of r when 1/r^6 is only 1.0% of its value at r=1 in dipole-dipole interactions is approximately 4.64 Angstroms.
In summary, the ion-ion interaction has a longer range, with the force of interaction operating at around 100 Angstroms away from molecular contact. On the other hand, dipole-dipole interactions have a shorter range, with the force of interaction operating at approximately 4.64 Angstroms away from molecular contact.