What is the minimum amount of work that must be done by the cell to restore V_m to -70 mV?

(a) 3 mJ;

(b) 3 µJ;

(c) 3 nJ;
(d) 3 pJ.

Upon fertilization, the eggs of many species undergo a rapid change in potential difference across their outer membrane. This change affects the physiological development of the eggs. The potential difference across the membrane is called the membrane potential, V_m , which is the potential inside the membrane minus the potential outside it. The membrane potential arises when enzymes use the energy available in ATP to expel three sodium ions (Na⁺) actively and accumulate two potassium ions (K⁺) inside the membrane—making the interior less positively charged than the exterior. For a sea urchin egg, V_m is about -70 mV; that is, the potential inside is 70 mV less than that outside. The egg membrane behaves as a capacitor with a capacitance of about 1 µF/cm². The membrane of the unfertilized egg is selectively permeable to K+; that is, K+ can readily pass through certain channels in the membrane, but other ions cannot. When a sea urchin egg is fertilized, Na⁺ channels in the membrane open, Na⁺ enters the egg, and V_m rapidly increases to +30 mV, where it remains for several minutes. The concentration of Na⁺ is about 30 mmol/L in the egg’s interior but 450 mmol/L in the surrounding seawater. The K⁺ concentration is about 200 mmol/L inside but 10 mmol/L outside. A useful constant that connects electrical and chemical units is the Faraday number, which has a value of approximately 10⁵ C/mol; that is, Avogadro’s number (a mole) of monovalent ions, such as Na⁺ or K⁺, carries a charge of 10⁵ C.