41. The spine of a neuron is approximately a sphere of diameter 1μm. The
concentration of K+ within this spine is 150mM. How many K+ ions are in this
sphere? Assume Avagadro’s number is 6.023 * 1023. Choose the option closest
to the correct answer.
a. 5 × 1010
b. 5 × 107
c. 5 × 105
d. 5 × 1012
The spine contains approximately 4.73 × 107 K+ ions, so option b. 5 × 107 is closest.
Step-by-Step Calculation
Radius r = 0.5 μm = 0.5 × 10-6 m.
Volume V = (4/3)πr3 = 5.236 × 10-19 m3 = 5.236 × 10-16 L (since 1 L = 10-3 m3).
Concentration 150 mM = 0.15 mol/L yields moles of K+ = 0.15 × 5.236 × 10-16 = 7.854 × 10-17 mol. Multiply by Avogadro’s number (6.023 × 1023) gives 4.73 × 107 ions.
This matches typical intracellular K+ levels in neurons around 150 mM.
Option Analysis
a. 5 × 1010: Overestimates by ~1000-fold; assumes ~10-13 L volume (1000 μm sphere), not 1 μm. Wrong scale for spine.
b. 5 × 107: Closest to 4.73 × 107; correct for ~0.52 μm³ volume at 150 mM. Best match.
c. 5 × 105: Underestimates by ~100-fold; like 1.5 mM or 10-18 L volume. Too low for spine.
d. 5 × 101: Severely underestimates; only 50 ions, as if 10-22 L or μM concentration. Irrelevant.
