Red Blood Cell Tonicity Lab (2D): Boyle-van 't Hoff, Hemolysis and Permeating Solutes

Osmosis and the Boyle-van 't Hoff law

Water moves towards the more concentrated side. Only the water part of the cell responds: V = V_b + V_w, where V_b (about 30 % of V₀, a typical figure) is osmotically inactive (hemoglobin and membrane). In an impermeant bath of osmolarity π the equilibrium is V = V_b + n_i/π, a straight line against 1/π, where n_i = 0.29 Osm/L × V_w0 is the fixed amount of impermeant solute inside.

Kinetics (Kedem-Katchalsky form)

dV_w/dt = L_w [ (n_i + σ n_s)/V_w − (c_np + σ c_p) ] and dn_s/dt = P_s (c_p − n_s/V_w). The reflection coefficient σ < 1 means a solute that leaks exerts less osmotic force. When the permeant solute has equilibrated the cell settles at V_w = n_i/c_np; with c_np = 0 (pure urea) it never stops swelling.

Tonicity is not osmolarity

290 mOsm/L of urea has the osmolarity of plasma but is hypotonic to the cell, because urea crosses the membrane and cannot hold water back.

Shape and bursting

Membrane area is nearly constant, so a biconcave disc has about 64 % spare volume before it is a sphere (V/V₀ = 1.64 for the equal-area sphere used here). Membranes rupture after a few percent area stretch, at a critical V/V₀ of about 1.6 with a cell-to-cell spread of 0.07, which gives the S-shaped osmotic fragility curve: hemolysis = Φ((V_max/V₀ − 1.6)/0.07). The 50 % point for an impermeant bath is c₅₀ = n_i/(1.6 V₀ − V_b). The side view uses the Evans-Fung biconcave profile (C₀, C₁, C₂ = 0.2072, 2.0026, −1.1228) blended towards a sphere.

Honest simplifications

• One mean cell is integrated; the 80 drawn cells differ only in their burst threshold (spread over the Gaussian quantiles).
• L_w and P_s are illustrative (real water exchange takes about 0.1 s; it is slowed here so it can be watched).
• No cytoskeleton mechanics, ion pumps, Donnan effects, pH, membrane-area change or volume-regulating channels. Drawn sizes are not to scale. Educational model, not clinical advice.

• Press "Hypotonic 0.5 % NaCl (160)": the cells swell to a sphere and about half burst, near the analytic c50.
• Press "Isotonic urea": the cells first shrink, then swell and burst because urea carries water in behind it.
• Lower the reflection coefficient to 0.5 and repeat the half-saline + permeant preset.
• Drag the impermeant salt slider and watch the live dot on the Boyle-van 't Hoff graph and the vertical marker on the fragility curve.
• Raise the inactive fraction to 0.45: the cells become less responsive and the BvH line gets a higher intercept.