Live practice

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

For a conjugate acid-base pair at 298 K, KaKb=Kw=1.0×10⁻¹⁴. Divide Kw by the acid ionisation constant in each case.

For charge −1, electron count e=Z+1. Use N=1.111e and A=N+Z.

If \(A\subset B\), no element of \(A\) lies outside \(B\), hence \(A-B=\varnothing\). If \(A-B=\varnothing\), every element of \(A\) lies in \(B\), so adding \(A\) to \(B\) gives \(A\cup B=B\). If \(A\cup B=B\), every element of \(A\) belongs to \(B\); hence the elements common to \(A\) and \(B\) are exactly those of \(A\), so \(A\cap B=A\). Finally, \(A\cap B=A\) means every element of \(A\) is in \(B\), giving \(A\subset B\). The implications form a closed chain, so all four conditions are equivalent.

Name the inner functions, differentiate each factor, and then simplify.

Substitution into the original expression verifies the stated domain and final value.

Translate every condition into an equation and retain exact values throughout the ellipse calculation.

The last line satisfies the original conditions and supplies every requested part.

Translate every condition into an equation and retain exact values throughout the straight-line calculation.

The last line satisfies the original conditions and supplies every requested part.

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.

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

For phenol alone, let x=[phenolate] and use Ka≈x²/C. With sodium phenolate present, the common ion fixes [phenolate]≈0.010 M and suppresses ionisation.