Question

In solid sodium chloride (table salt), chloride ions have one more electron than they have protons, and sodium ions have one more proton than they have electrons. These ions are separated by about \(0.28 \mathrm{nm} .\) Calculate the electrostatic force between a sodium ion and a chloride ion.

Short answer

Answer: The electrostatic force between a sodium ion and a chloride ion in solid sodium chloride is approximately \(2.92 \times 10^{-9} \,\mathrm{N}\).

Step-by-step solution

Gather Given Information

The given information from the problem includes: - The distance between a sodium ion and a chloride ion is \(0.28 \mathrm{nm}\). - Sodium ions have one more proton than electrons. - Chloride ions have one more electron than protons. We also know that the charge of an electron is \(q_e = -1.6 \times 10^{-19} \mathrm{C}\) and the charge of a proton is \(q_p = 1.6 \times 10^{-19} \mathrm{C}\).

Calculate the Charges of Sodium and Chloride Ions

Since sodium ions have one more proton than electrons, its charge is \(q_{Na} = q_p\). For chloride ions, since they have one more electron than protons, its charge is \(q_{Cl} = q_e\). Therefore, we have: - Charge of sodium ion (Na+): \(q_{Na} = 1.6 \times 10^{-19} \mathrm C\) - Charge of chloride ion (Cl-): \(q_{Cl} = -1.6 \times 10^{-19} \mathrm C\)

Convert Distance from nanometers to meters

The distance between the ions is given in nanometers (\(\mathrm{nm}\)). We need to convert this to meters (\(\mathrm{m}\)) for using it in the calculation. 1 nm = \(10^{-9}\) meters So, \(0.28 \mathrm{nm} = 0.28 \times 10^{-9} \,\mathrm{m} = 2.8 \times 10^{-10} \mathrm{m}\).

Calculate the Electrostatic Force using Coulomb's Law

Coulomb's Law states that the electrostatic force F between two charges of q1 and q2 separated by a distance r is given by: \(F = k \frac{q1q2}{r^2} \) Where k is the electrostatic constant, \(k = 8.99\times10^{9} \mathrm{Nm}^2/\mathrm{C}^2\). Now, we can apply Coulomb's Law to find the electrostatic force between the sodium and chloride ions. \(F = 8.99\times 10^{9} \displaystyle\frac{(1.6\times 10^{-19})(-1.6\times 10^{-19})} {(2.8\times 10^{-10})^{2}}\)

Simplify and Calculate the Electrostatic Force

Calculate the electrostatic force, F, using the above formula: \(F \approx 8.99\times10^{9} \displaystyle\frac{(2.56\times10^{-38})}{7.84\times10^{-20}}\) \(F \approx 2.92 \times 10^{-9}\,\mathrm{N}\) So, the electrostatic force between a sodium ion and a chloride ion in solid sodium chloride is approximately \(2.92 \times 10^{-9} \,\mathrm{N}\).

Key concepts

Coulomb's Law

Coulomb's Law is fundamental when it comes to understanding the attractive or repulsive force between two charged particles. It's described by the simple equation:
\[ F = k \frac{q1 \cdot q2}{r^2} \]
where:

• \(F\) is the electrostatic force between the charges
• \(q1\) and \(q2\) are the magnitudes of the charges
• \(r\) is the distance between the centers of the two charges
• \(k\) is the electrostatic constant

The law demonstrates that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This inverse square law is a pervasive concept in physics, seen in other fundamental forces as well. For instance, the larger the charges or the closer they are together, the stronger the force they'll exert on each other.

Charge of Ions

In ionic compounds like sodium chloride, we deal with ions – atoms or molecules that have lost or gained electrons and thus have a net charge. The charge of an ion is quantified in terms of elementary charge units, where one elementary charge (commonly denoted as \(e\)) is approximately \(1.6 \times 10^{-19} \mathrm{C}\) (coulombs).
In our example, a sodium ion, represented as Na+, has lost one electron and thus carries a positive charge of \(+1e\) or \(+1.6 \times 10^{-19} \mathrm{C}\). Conversely, a chloride ion, represented as Cl-, has gained an electron, resulting in a negative charge of \(-1e\) or \(-1.6 \times 10^{-19} \mathrm{C}\). The reason for these charges is due to the opposite nature of protons (positive) and electrons (negative). Understanding ion charges helps us predict the behavior of ions in terms of attraction and repulsion.

Electrostatic Constant

In Coulomb's Law, the constant \(k\), also known as the electrostatic constant, Coulomb's constant, or electric force constant, is a proportionality factor that helps us calculate the force of interaction between two charged particles. The accepted value for this constant is approximately \(8.99 \times 10^9 \mathrm{N m^2/C^2}\).
This constant arises from the electric permittivity of free space, which defines how electric fields interact with the physical vacuum. When using this constant, it's important to maintain consistent units throughout your calculations to ensure accuracy, which often involves converting distances to meters and charges to coulombs. The electrostatic constant is one of those values you'll use frequently in physics problems dealing with charges.

Unit Conversion

When solving physics problems, especially those involving electrostatics, you might need to convert units to maintain consistency. In the solution above, we converted the distance between ions from nanometers to meters because Coulomb's Law requires that we use the System International (SI) unit of meters for distance.
Here’s an example of such a conversion:
\[1 \mathrm{nm} = 10^{-9} \mathrm{m}\]
So, for our distance of \(0.28 \mathrm{nm}\), this translates to:
\[0.28 \mathrm{nm} = 0.28 \times 10^{-9} \mathrm{m} = 2.8 \times 10^{-10} \mathrm{m}\]
Similar conversions may be needed for other units, such as converting microcoulombs to coulombs by recognizing that \(1 \mu\mathrm{C} = 10^{-6} \mathrm{C}\). It's crucial to master unit conversions to ensure the accuracy of calculations in physics.