In Osmosis and Tonicity you predicted where water goes from solute concentrations. Plants add a twist: a plant cell has a wall, and pressure inside it pushes water back out. Water potential (Ψ, the Greek letter psi) combines both effects into one number. You will need molarity from chemistry.
Key idea
$$\Psi = \Psi_s + \Psi_p \qquad \Psi_s = -iCRT$$ Water moves from higher water potential to lower water potential. Pure water at normal air pressure has $\Psi = 0$. Solutes make $\Psi_s$ negative; pressure (turgor) makes $\Psi_p$ positive.
- $i$: the ionization constant, the particles each formula unit makes in water (sucrose 1, NaCl 2, CaCl₂ 3).
- $C$: molar concentration (mol/L).
- $R = 0.0831$ L·bar/(mol·K), the pressure constant.
- $T$: temperature in kelvin (°C + 273).
- Water potential is measured in bars (1 bar is about 1 atmosphere).
Why it works
Why does solute lower the water potential? Water molecules that are busy surrounding solute particles are less free to move. So pure water, with nothing holding it back, has the most free energy, and any solution has less: a negative $\Psi_s$. Water flows "downhill" toward lower Ψ, which means toward more solute, unless pressure pushes back. In a plant cell, water enters until the wall presses hard enough that $\Psi_p$ cancels the difference.
Worked examples
Worked example: Solute potential of sucrose
Find the solute potential of a 0.1 M sucrose solution at 22 °C.
- Sucrose does not ionize: $i = 1$. $T = 22 + 273 = 295$ K.
- $\Psi_s = -iCRT = -(1)(0.1)(0.0831)(295)$
- $= -2.45$ bar.
Answer: $\Psi_s \approx -2.45$ bar. In an open beaker, $\Psi_p = 0$, so $\Psi = -2.45$ bar too.
Worked example: Which way does water go?
A plant cell has $\Psi_s = -7.0$ bar and $\Psi_p = +3.0$ bar. It sits in 0.2 M NaCl at 22 °C. Does water enter or leave the cell?
- Cell: $\Psi = -7.0 + 3.0 = -4.0$ bar.
- Solution: NaCl gives 2 ions, so $i = 2$. $\Psi_s = -(2)(0.2)(0.0831)(295) = -9.81$ bar, and $\Psi_p = 0$.
- $-4.0$ is higher than $-9.81$.
Answer: Water moves from the cell (−4.0 bar) to the solution (-9.8 bar): the cell loses water.
Common mistake
Two slips. Thinking water moves toward higher water potential: it goes from high to low, like a ball rolling downhill. And dropping the minus sign in $\Psi_s = -iCRT$, which makes a salty solution look like it has more water potential than pure water. Check: any solution must come out with $\Psi_s$ below 0, and more solute must give a more negative number.
In the classic potato lab, cores sit in sucrose solutions of different molarity. Where the line of percent mass change crosses zero, the solution is isotonic with the potato, and the potato's Ψ equals the Ψ of that solution.