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Describe the change at Al when AlCl₃ accepts Cl⁻ to form AlCl₄⁻.
Al in trigonal-planar AlCl₃ has three electron domains and sp² hybridisation. Accepting a lone pair gives four Al–Cl sigma bonds in a tetrahedral ion.
Al changes from sp² in AlCl₃ to sp³ in AlCl₄⁻.
Should two isotopes of one element have the same first ionisation enthalpy? Justify.
Isotopes have the same nuclear charge and the same ground-state electronic configuration. Their mass difference has a negligible influence on electronic binding at this level.
Their first ionisation enthalpies are effectively the same.
For an electron in a 3d orbital, list the allowed n, l and mₗ values.
The 3d label fixes n=3 and l=2; mₗ takes every integral value from −l to +l.
Explain the difference between 0.50 mol Na₂CO₃ and a 0.50 M Na₂CO₃ solution.
The mole is an amount of substance, whereas molarity states the amount dissolved per litre of final solution.
Evaluate the finite sum from k=1 to 11 of (2+3ᵏ).
Use the sequence definition or geometric-progression formula, keeping the first-term offset explicit.
The displayed substitutions cover every requested part and verify the result against the original data.
How many chords are determined by 21 points on a circle?
Every unordered pair of points determines one chord.
A triangle’s longest side is three times its shortest side; the third side is \(2\,\mathrm{cm}\) shorter than the longest. If its perimeter is at least \(61\,\mathrm{cm}\), find the minimum shortest side.
Let the shortest side be x cm. The other sides are 3x and 3x−2.
Let \(f(x)=x^2\) for \(0\le x\le3\) and \(f(x)=3x\) for \(3\le x\le10\). Let \(g(x)=x^2\) for \(0\le x\le2\) and \(g(x)=3x\) for \(2\le x\le10\). Show that \(f\) is a function but \(g\) is not.
A piecewise rule is a function only if overlapping rules give the same value. At the common point \(x=3\), \(3^2=9\) and \(3(3)=9\); therefore \(f(3)\) is unique. At \(x=2\), however, the rules for \(g\) give \(2^2=4\) and \(3(2)=6\). One input has two images, so \(g\) is not a function.
Find the intersections of the five pairs used in Exercise 1.4, Question 1.
\(\{1,3\},\{a\},\{3\},\varnothing,\varnothing\).
Which expression does not represent electric power?
Valid forms are \[\begin{aligned}P &= VI\\&= I^2R\\&= \frac{V^2}{R}\end{aligned}\]. \(IR^2\) has incorrect dimensions.
\(IR^2\) - option (B).