Problem 18 - Chemistry: The Central Science

Question

Arrange the following atoms in order of increasing effective nuclear charge experienced by the electrons in the \(n=3\) electron shell: K, Mg, P, Rh, Ti.

Step-by-Step Solution

Answer

The order of increasing effective nuclear charge experienced by the electrons in the \(n=3\) electron shell for the given elements is: Mg, P, K, Ti, Rh.

Step 1: Identify the atomic number of each element

Look up the atomic number (Z) for each element, which is the number of protons in the nucleus: - K (Potassium): Z = 19 - Mg (Magnesium): Z = 12 - P (Phosphorus): Z = 15 - Rh (Rhodium): Z = 45 - Ti (Titanium): Z = 22

Step 2: Calculate shielding constant for each element

For electron shells with principal quantum number n=3, the shielding constant (S) can be roughly estimated using Slater's rules as follows: 1. S = 0.85 for each electron in the same shell (n=3) 2. S = 1 for each electron in the (n - 1) shell 3. S = 0.35 for each electron in the (n - 2) shell By applying these rules to each element, we get the following shielding constants (S): - K: S = 1(8) + 0.35(2) = 9.7 - Mg: S = 1(8) + 0.35(2) = 9.7 - P: S = 1(8) + 0.35(2) + 0.85(3) = 11.05 - Rh: S = 1(18) + 0.35(28) + 0.85(3) = 27.8 - Ti: S = 1(8) + 0.35(2) + 0.85(2) = 10.2

Step 3: Calculate the effective nuclear charge (Z_eff) for each element

Use the formula Z_eff = Z - S to calculate the effective nuclear charge for each element: - K: Z_eff = 19 - 9.7 = 9.3 - Mg: Z_eff = 12 - 9.7 = 2.3 - P: Z_eff = 15 - 11.05 = 3.95 - Rh: Z_eff = 45 - 27.8 = 17.2 - Ti: Z_eff = 22 - 10.2 = 11.8

Step 4: Arrange elements in order of increasing effective nuclear charge

Now that we have calculated the effective nuclear charge for each element, we can arrange them in order of increasing Z_eff: 2.3 (Mg) < 3.95 (P) < 9.3 (K) < 11.8 (Ti) < 17.2 (Rh) Hence, the order of increasing effective nuclear charge experienced by the electrons in the n=3 electron shell for the given elements is: Mg, P, K, Ti, Rh.

Key Concepts

Periodic Trends in Atomic Properties

Understanding the trends in atomic properties across the periodic table is crucial for predicting the behavior of atoms. As you move across a period (from left to right), the effective nuclear charge experienced by electrons generally increases. This is because each successive element has an additional proton, increasing the positive charge in the nucleus. Meanwhile, as you move down a group, the principal quantum number increases, adding electron shells that can shield the inner electrons from the nucleus's pull, leading to a decrease in effective nuclear charge felt by these inner electrons.

Impact on Atomic Size and Reactivity

These trends have significant implications. For example, increasing effective nuclear charge across a period leads to a decrease in atomic radius, while an increase down a group results in larger atomic sizes. Also, elements with higher effective nuclear charges tend to hold onto their electrons more tightly, affecting their chemical reactivity.

Slater's Rules

Slater's rules provide a method to estimate the shielding effect experienced by an electron in a multi-electron atom. It takes into account the repulsion from other electrons when determining the net positive charge (effective nuclear charge) felt by a specific electron. According to Slater's rules, electrons are grouped into 'shells' and 'subshells' based on the principal quantum number and the type of orbital.

The shielding effect for a given electron is then calculated by adding up contributions from electrons in the same shell, which shield less effectively, and those from inner shells, which shield more effectively. Using Slater's rules, you can assess the effect of electron shielding on atomic properties such as ionization energy and electron affinity, which are essential for understanding chemical behavior.

Atomic Number

The atomic number, denoted as Z, is the number of protons in an atom's nucleus and serves as a fundamental identifier for elements. It is unique to each element and determines its position on the periodic table. As the atomic number increases, so does the number of electrons in a neutral atom, which impacts the effective nuclear charge. In textbook exercises, like the one provided, knowing the atomic number is the starting point for calculating many properties, including the effective nuclear charge.

Shielding Effect

The shielding effect describes the phenomenon where inner electrons reduce the attractive force exerted by the nucleus on the outer electrons. This effect becomes more significant with atoms that have multiple electron shells. Inner-shell electrons repulse outer-shell electrons, leading to a decrease in the net nuclear charge that these outer electrons experience. The shielding effect is a critical concept when predicting an atom's reactivity, size, and how tightly it holds on to its electrons. Calculations often rely on approximate methods, such as Slater's rules, to determine the magnitude of this effect.

Principal Quantum Number

The principal quantum number, symbolized as n, defines the electron shell of an atom and correlates with an electron's energy level and distance from the nucleus. As n increases, the average distance of an electron from the nucleus also increases. It also indicates the number of subshells within a shell, affecting the electron configuration of an atom. Understanding the principal quantum number is essential when applying Slater's rules to estimate the effective nuclear charge, as seen in the step-by-step solution provided in the exercise. The principal quantum number helps us to determine the electron's location which in turn influences the shielding constant used in those calculations.