Biological importance of Na, K, Ca, Mg: Worked examples

This page is the drill ground for the parent topic. The parent gave you the ideas; here we throw every kind of question at you — every sign of voltage, every degenerate case (zero gradient, equal concentrations), the limits, a word problem, and an exam trap.

Before any number: the one engine we lean on is the Nernst equation. Let us re-earn each symbol so a first-timer can follow.

Definition Re-reading the Nernst equation from zero

$E = \frac{R T}{z F} ln \frac{[ X ] _{o u t}}{[ X ] _{in}}$

$E$ = the equilibrium voltage across the membrane, in volts. It is the voltage at which the ion stops moving because the electric pull and the concentration pull cancel.
$R$ = the gas constant $= 8.314 J mol^{- 1} K^{- 1}$ . It converts "temperature" into "energy per mole".
$T$ = absolute temperature in kelvin. Body temperature $37^{\circ} C = 310 K$ .
$z$ = the charge of the ion with its sign ( $+ 1$ for Na⁺/K⁺, $+ 2$ for Ca²⁺/Mg²⁺).
$F$ = the Faraday constant $= 96485 C mol^{- 1}$ — the charge carried by one mole of $+ 1$ ions.
$ln$ = natural logarithm — it appears because chemical work of moving particles down a gradient grows with the log of the concentration ratio, not the ratio itself.
$[X]_{o u t}, [X]_{in}$ = concentration of ion outside / inside the cell (any consistent unit — the ratio is what counts, so mM/mM is fine).

The scenario matrix

Every question this topic can ask falls into one of these case classes. The examples underneath are tagged with the cell they cover.

#	Case class	What is special	Covered by
C1	$z = + 1$ , ion higher outside (Na⁺)	ratio $> 1$ → positive $E$	Ex 1
C2	$z = + 1$ , ion higher inside (K⁺)	ratio $< 1$ → negative $E$	Ex 2
C3	$z = + 2$ ion (Ca²⁺)	dividing by $z = 2$ halves the voltage	Ex 3
C4	Degenerate: equal concentrations	ratio $= 1$ , $ln 1 = 0$ → $E = 0$	Ex 4
C5	Limit: extreme gradient ( $1 0^{4}$ -fold)	how big can $E$ get?	Ex 5
C6	Scaling invariance (double both sides)	tests the "ratio only" fact	Ex 6
C7	Word problem — pump energy budget	stoichiometry: 3 Na out, 2 K in per ATP	Ex 7
C8	Word problem — bone chemistry	mass of Ca from hydroxyapatite	Ex 8
C9	Exam twist — spot the wrong metal	Mg vs Fe in porphyrin	Ex 9

Worked examples

Worked example Ex 1 — C1: Sodium's equilibrium voltage (positive)

Find $E_{N a}$ at body temperature with $[N a]_{o u t} = 145 mM$ , $[N a]_{in} = 12 mM$ .

Forecast: Na⁺ is much richer outside. It "wants" to rush in. To stop it, the inside must be made positive to repel it — so guess $E_{N a}$ is a positive number, tens of millivolts.

Identify $z = + 1$ (Na⁺ carries one positive charge). Why this step? $z$ sits in the denominator; getting its sign and size right fixes both magnitude and sign of the whole answer.
Form the ratio $\frac{[ N a ] _{o u t}}{[ N a ] _{in}} = \frac{145}{12} = 12.08$ . Why this step? Nernst only sees the ratio; $12.08 > 1$ already tells us the log is positive.
Plug into the shortcut: $E_{N a} = \frac{61.5}{1} lo g_{10} (12.08)$ . Why this step? Using the $61.5/ z$ form skips the messy $R T / F$ arithmetic.
$lo g_{10} (12.08) = 1.082$ , so $E_{N a} = 61.5 \times 1.082 \approx + 66.5 mV$ .

Verify: Sign is $+$ as forecast (ion higher outside → positive $E$ ). Physiology textbooks quote $E_{N a} \approx + 60 mV$ — same ballpark. ✓

Worked example Ex 2 — C2: Potassium's equilibrium voltage (negative)

Find $E_{K}$ with $[K]_{o u t} = 4 mM$ , $[K]_{in} = 140 mM$ .

Forecast: K⁺ is richer inside, so it leaks out. To hold it in, the inside must go negative (to attract the positive K⁺ back). Guess: a negative voltage near the $- 70 mV$ resting value.

$z = + 1$ .
Ratio $= \frac{4}{140} = 0.02857$ . Why this step? Ratio $< 1$ guarantees a negative log — the source of the minus sign.
$lo g_{10} (0.02857) = - 1.544$ . Why this step? Logs of numbers below 1 are negative; this is the whole reason $E_{K}$ is negative.
$E_{K} = 61.5 \times (- 1.544) \approx - 95 mV$ .

Verify: Negative, as forecast, and close to the resting $- 70 mV$ — which is exactly why the parent note says K⁺ mainly sets the resting potential. ✓ See the figure below for how the two cases mirror each other.

Figure — Biological importance of Na, K, Ca, Mg

Worked example Ex 3 — C3: Calcium — the divisor

$z = 2$ bites Find $E_{C a}$ with $[C a]_{o u t} = 1 0^{- 3} M$ , $[C a]_{in} = 1 0^{- 7} M$ .

Forecast: A colossal $1 0^{4}$ -fold gradient (Ca is $10, 000 \times$ richer outside). You might expect a huge positive voltage — but Ca²⁺ has $z = 2$ , and dividing by 2 halves it. Guess: large and positive, but roughly half of what a $z = 1$ ion would give.

Ratio $= \frac{1 0 ^{- 3}}{1 0 ^{- 7}} = 1 0^{4}$ . Why this step? Keeping powers of ten makes the log trivial.
$lo g_{10} (1 0^{4}) = 4$ exactly.
$E_{C a} = \frac{61.5}{2} \times 4 = 30.75 \times 4 = + 123 mV$ . Why the $z = 2$ ? Each Ca²⁺ carries two charges, so the electric force does double the work per ion — the concentration force needs only half the voltage per unit log to balance it.

Verify: If Ca had been $z = 1$ it would give $61.5 \times 4 = 246 mV$ ; the divalent value is exactly half, $123 mV$ . Positive and large as forecast. ✓

Worked example Ex 7 — C7: Word problem — the pump's charge budget

The Na⁺/K⁺-ATPase burns 1 ATP to push 3 Na⁺ out and pull 2 K⁺ in. (a) What is the net charge moved across the membrane per cycle? (b) If a cell runs $6 \times 1 0^{6}$ pump cycles per second, how many Na⁺ ions leave per second?

Forecast: More positive charge leaves (3) than enters (2), so the outside should get more positive — a net of $+ 1$ charge out per cycle. And Na⁺ out per second is just $3 \times$ the cycle rate.

Net charge out $= (3 \times + 1) - (2 \times + 1) = + 3 - 2 = + 1$ per cycle. Why this step? Only the difference leaves a lasting voltage; matched charges cancel.
This is why the pump is called electrogenic — it directly makes the inside slightly more negative.
Na⁺ per second $= 3 \times 6 \times 1 0^{6} = 1.8 \times 1 0^{7}$ ions s⁻¹. Why this step? 3 Na⁺ leave each of the $6 \times 1 0^{6}$ cycles.

Verify: Net $+ 1$ charge out per cycle ✓ (matches the known electrogenic behaviour). $3 \times 6 \times 1 0^{6} = 1.8 \times 1 0^{7}$ ✓ — units: (ions/cycle) $\times$ (cycles/s) = ions/s. ✓

Worked example Ex 8 — C8: Word problem — calcium locked in bone

Hydroxyapatite is $Ca_{10} (PO_{4})_{6} (OH)_{2}$ , molar mass $\approx 1004 g mol^{- 1}$ . How many grams of calcium element are in one mole of hydroxyapatite? (Take Ca $= 40 g mol^{- 1}$ .)

Forecast: There are 10 Ca atoms in the formula, each $40 g mol^{- 1}$ , so guess $400 g$ — meaning calcium is roughly 40 % of the mineral's mass.

Ca atoms per formula unit $= 10$ (the subscript on Ca). Why this step? The subscript literally counts atoms per formula — the structure here is stoichiometry, not geometry.
Mass of Ca $= 10 \times 40 = 400 g$ per mole of hydroxyapatite.
Mass fraction $= \frac{400}{1004} = 0.398 = 39.8%$ . Why this step? Fraction tells you how "calcium-dense" bone mineral is.

Verify: $10 \times 40 = 400 g$ ✓; $400/1004 \approx 0.398$ , i.e. $\approx 40%$ — consistent with the forecast and with why bone is the body's calcium bank (99 % of body Ca). ✓

Worked example Ex 9 — C9: Exam twist — spot the wrong metal

An exam claims: "Chlorophyll and haemoglobin both use a porphyrin ring holding an iron ion at the centre." Identify and fix the error, then state what actually determines the colour job.

Forecast: The porphyrin frame is shared, but the central metal differs. Guess: chlorophyll uses Mg²⁺, not iron.

Recall the pairing: chlorophyll $\to$ Mg²⁺; haemoglobin $\to$ Fe²⁺. Why this step? Both sit in the same porphyrin cage, so the only discriminator is the metal — that is precisely the trap.
Fix the statement: "…holding Mg²⁺ in chlorophyll and Fe²⁺ in haemoglobin."
Why it matters: the central metal tunes which light is absorbed — Mg-porphyrin absorbs red/blue and reflects green, driving photosynthesis.

Verify: Matches the parent's [!mistake] box ("Iron is in chlorophyll" is false). Same frame, different metal. ✓

Recall Quick self-test

Sign of $E$ when an ion is richer outside ( $z = + 1$ )? ::: Positive (like Na⁺, $\approx + 66 mV$ ) Sign of $E$ when an ion is richer inside? ::: Negative (like K⁺, $\approx - 95 mV$ ) Equal concentrations inside and out give $E =$ ? ::: $0 mV$ (ratio 1, $lo g 1 = 0$ ) Why is Ca²⁺'s $123 mV$ only half of a $z = 1$ value at the same $1 0^{4}$ ratio? ::: Because $z = 2$ divides the voltage Doubling both concentrations changes $E$ by? ::: Nothing — only the ratio matters Net charge moved out per Na⁺/K⁺-ATPase cycle? ::: $+ 1$ (3 out $-$ 2 in) Central metal of chlorophyll vs haemoglobin? ::: Mg²⁺ vs Fe²⁺

Connections

Nernst Equation — the single engine behind Ex 1–6
Alkali Metals (Na, K) — why Na⁺/K⁺ are $z = + 1$ and mobile
Alkaline Earth Metals (Mg, Ca) — why Ca²⁺/Mg²⁺ are $z = + 2$
ATP and Bioenergetics — pump energy budget (Ex 7)
Chlorophyll and Photosynthesis · Coordination Chemistry — Porphyrins — Mg vs Fe (Ex 9)
Osmosis and Fluid Balance — Na⁺ gradients and blood pressure