Question

If the extracellular concentration of Na$^+$ is ten times its intracellular concentration, then the sodium equilibrium potential at $20^\circ$C in mV is __________ (rounded off to two decimal places). Assume the membrane is permeable only to Na$^+$ ions. [Use $R=1.987\ \mathrm{cal\,deg^{-1}\,mol^{-1}}$, $F=23062\ \mathrm{cal\,mol^{-1}\,V^{-1}}$]}

Hint:

At room temperature, a $10\times$ gradient for a monovalent ion gives $\approx 58\ \mathrm{mV}$; at body temperature ($37^\circ$C), remember $\approx 61.5\ \mathrm{mV}$.

Answer 1

Step 1: Nernst equation.
For monovalent cation ($z=+1$): $E = \dfrac{RT}{zF}\ln\!\left(\dfrac{[\mathrm{Na^+}]_\text{out}}{[\mathrm{Na^+}]_\text{in}}\right)$
Step 2: Substitute values at $T=293\ \mathrm{K$ (20$^\circ$C).}
$\dfrac{RT}{F}=\dfrac{(1.987)(293)}{23062}=0.025245\ \mathrm{V}$
Ratio $=10 ⇒ \ln 10 = 2.302585$
$E = 0.025245 \times 2.302585 = 0.05812\ \mathrm{V} = 58.12\ \mathrm{mV}$.
Hence, the Na$^+$ equilibrium potential is about $58\ \mathrm{mV$ (inside negative)} when outside is $10\times$ inside.