Potassium has the electronic configuration [Ar] 4s1, so the outermost electron is in n=4.

The ground state of hydrogen corresponds to n=1.

Potassium has an atomic number of 19, and its outermost electrons are in the n=4 shell.

Sodium has an atomic number of 11, and its outermost electrons are in the n=3 shell.

The series diverges to infinity as n approaches infinity.

The universal gas constant R is 8.314 J/(mol·K).

The universal gas constant R is 0.0821 L·atm/(K·mol).

Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

Solving for x: 3x = 21 => x = 7.

Using the quadratic formula, x = [8 ± sqrt(64 - 48)] / 4 = [8 ± 4] / 4, giving x = 3 or x = 1.

Solving for x: 3x = 12, thus x = 4.

Expanding gives 5x - 10 = 3x + 4. Rearranging gives 2x = 14, thus x = 7.

Rearranging gives 5x - 2x = 12 + 3 => 3x = 15 => x = 5.

Rearranging gives 5x - 2x = 6 + 3 => 3x = 9 => x = 3.

The solutions to z^2 = -1 are z = i and z = -i.

The determinant is 0 because the first column is repeated.

By definition, \( \tan(\tan^{-1}(x)) = x \). Therefore, \( \tan(\tan^{-1}(3)) = 3 \).

|z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.

|z| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.

The modulus |z| = 4, as it is the coefficient in the polar form.

|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.

NaCl dissociates into two ions (Na+ and Cl-), so the van 't Hoff factor (i) is 2.

The van 't Hoff factor (i) is equal to the number of particles the solute dissociates into. For a strong electrolyte that dissociates into 3 ions, i = 3.

The van 't Hoff factor (i) for glucose is 1, as it does not dissociate in solution.

For a non-electrolyte solute, the van 't Hoff factor (i) is 1, as it does not dissociate into ions.

Vapor pressure = P°_solvent * (n_solvent / (n_solvent + n_solute)) = 23.76 * (55.5 / (55.5 + 0.1)) = 22.88 mmHg

Using Raoult's law, the vapor pressure of the solution = (moles of solvent / total moles) * P0 = (10 / 11) * P0 = 0.909 P0.

Vapor pressure lowering = (moles of solute / total moles) * P0 = (1 / (1 + 3)) * P0 = 0.25 P0, so vapor pressure = P0 - 0.25 P0 = 0.75 P0.

Using Raoult's law, P_solution = X_solvent * P°_solvent = (9/10) * 100 mmHg = 90 mmHg.

Using Raoult's law, P_solution = X_solvent * P°_solvent. X_solvent = 3/(1+3) = 0.75, so P_solution = 0.75 * 100 mmHg = 75 mmHg.