Electrochemistry and circuits

The inverse-square distance dependence means doubling the separation quarters the force. The constant k ≈ 9×10⁹ N·m²/C² is large, so even small charges exert big forces at short range. The parallel to Newtonian gravity is exact in form (F = Gm₁m₂/r²), a frequent MCAT analogy — but gravity is only attractive, while the electrostatic force has a sign.

A point can have zero field but nonzero potential (midway between two equal like charges: fields cancel, potentials add) or zero potential but nonzero field (midway between equal opposite charges: potentials cancel, fields add). Field lines run from high to low potential and are perpendicular to equipotential surfaces; no work is done moving a charge along an equipotential.

Ohm's law is the circuit workhorse — rearranged as I = V/R, it shows current rising with voltage and falling with resistance. The geometric formula R = ρL/A mirrors fluid resistance (long thin tubes resist more), a useful cross-section to the fluids chapter. Resistivity ρ is a material property; for most conductors it rises with temperature.

In series, removing any element breaks the whole circuit (old Christmas lights); in parallel, each branch is independent (household wiring). Adding a parallel resistor opens another path for current, so total resistance falls and total current rises — the counterintuitive result. Two equal resistors in parallel give half the resistance; in series, double.

Which form to use depends on what's held constant: at fixed current, power scales with R (I²R); at fixed voltage, power scales inversely with R (V²/R). This is why a short circuit (low R across a fixed voltage) dissipates huge power and overheats. Energy used is power times time (the kWh on a utility bill).

Capacitors store energy in the electric field between their plates. Inserting a dielectric raises capacitance (it reduces the field for the same charge) and lets the capacitor store more charge at a given voltage. Capacitors combine oppositely to resistors — parallel adds (C_total = ΣC), series gives a smaller total — a frequent trap when the circuit rules are applied by rote.

Kirchhoff's two rules solve circuits too complex for simple series/parallel reduction. The junction (node) rule follows from conservation of charge — current splits and recombines but is never lost. The loop rule follows from conservation of energy — voltage rises (across the battery) and drops (across resistors) net to zero around any closed loop. Placement matters: an ideal ammeter (zero resistance) goes in series, an ideal voltmeter (infinite resistance) goes in parallel.

The species oxidized loses electrons (its oxidation state rises) and is the reducing agent; the species reduced gains electrons (oxidation state falls) and is the oxidizing agent. Key assignment rules: a free element is 0; a monatomic ion equals its charge; oxygen is −2 (peroxides −1); hydrogen is +1 (metal hydrides −1); fluorine is always −1; states sum to the species' overall charge. Balance redox reactions with half-reactions — one oxidation, one reduction — balancing atoms and charge with electrons, then adding so the electrons cancel.

The constant across both cells is "An Ox, Red Cat" — anode = oxidation, cathode = reduction. What flips is the sign of the electrodes: in a galvanic cell the anode is negative and the cathode positive; in an electrolytic cell an external power supply makes the anode positive and the cathode negative. Electrons always flow from anode to cathode through the wire.

A more positive reduction potential = a stronger oxidizing agent (more eager to be reduced); the species with the lower (more negative) reduction potential is forced to oxidize and serves as the anode. Reduction potentials are intensive — they don't scale when you multiply a half-reaction, because they're energy per electron. The n in ΔG° = −nFE° is the number of moles of electrons transferred (F is Faraday's constant).

As a galvanic cell discharges, reactant concentrations fall and Q rises, so E declines toward zero (a dead battery is at equilibrium, E = 0). The Nernst equation also underlies the membrane potential of neurons, where ion concentration gradients set the resting voltage — a direct biology tie-in. In electrolysis, Faraday's laws convert charge to product: moles of electrons = It/F, then stoichiometry gives moles of metal deposited.