The atomic radii of a divalent cation and a monovalent anion are .50nm
 
 and .135nm
 
, respectively.
 Calculate the force of attraction between these two ions at their equilibrium interionic separation (i.e., when the ions just touch one another).
 What is the force of repulsion at this same separation distance?

The problem requires calculating both the **force of attraction** and the **force of repulsion** between a divalent cation and a monovalent anion at their equilibrium interionic separation, which is when the ions are just touching.

### 1. **Force of Attraction**

The force of attraction between two ions can be calculated using **Coulomb's law**:

\[
F_{\text{attraction}} = \frac{Z_1 Z_2 e^2}{4 \pi \varepsilon_0 r^2}
\]

Where:
- \( Z_1 \) and \( Z_2 \) are the charges on the ions (for the divalent cation, \( Z_1 = 2 \) and for the monovalent anion, \( Z_2 = 1 \)).
- \( e = 1.602 \times 10^{-19} \, \text{C} \) is the elementary charge.
- \( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \) is the permittivity of free space.
- \( r \) is the separation distance between the two ions. Since the ions are just touching, this is the sum of their ionic radii:

\[
r = 0.50 \, \text{nm} + 0.135 \, \text{nm} = 0.635 \, \text{nm} = 0.635 \times 10^{-9} \, \text{m}
\]

Now, let's calculate the force of attraction using these values.

### 2. **Force of Repulsion**

The force of repulsion is more complicated, as it is often modeled using an empirical formula of the form:

\[
F_{\text{repulsion}} = \frac{A}{r^n}
\]

Where \( A \) is a constant that depends on the nature of the ions, and \( n \) is a constant that describes how sharply the force of repulsion increases as the ions come closer. However, since no specific information about \( A \) or \( n \) is given in the problem, the exact force of repulsion cannot be calculated without additional data.

Let's proceed to calculate the force of attraction using Coulomb's law. I'll do this calculation next.The force of attraction between the divalent cation and the monovalent anion at their equilibrium interionic separation is approximately \( 1.144 \times 10^{-9} \, \text{N} \).

### For the force of repulsion:
Without specific values for the constants \( A \) and \( n \), the exact repulsive force cannot be calculated with the given information.

Would you like more details on this, or have any further questions?

Here are five related questions you might consider:
1. How does the charge of the ions affect the magnitude of the attractive force?
2. What happens to the force of attraction if the distance between the ions is doubled?
3. What assumptions are made in Coulomb's law about the medium between the ions?
4. How do you calculate the force of repulsion if the values of \( A \) and \( n \) are known?
5. How do these forces contribute to the stability of ionic compounds?

**Tip:** The attractive and repulsive forces between ions determine the structure and stability of ionic solids, such as salts.

The atomic radii of a divalent cation and a monovalent anion are 0.50 nm and 0.135 nm, respectively. Calculate the force of attraction between these two ions at their equilibrium interionic separation (i.e., when the ions just touch one another). What is the force of repulsion at this same separation distance?

This physics problem calculates the force of attraction between a divalent cation and a monovalent anion using Coulomb's Law. Explore how to find the force of attraction at equilibrium interionic separation and understand the empirical model for repulsive forces. Suitable for college-level physics students.

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