Free Water Potential Calculator with Worked Steps (Bars or MPa)

The two formulas, and which one your question is asking for

Water potential (Ψ, the Greek letter psi) measures how much free energy water has, and therefore which way it will move. Water always moves from higher water potential to lower water potential — from less negative to more negative. Pure water in an open container is the reference point and is defined as Ψ = 0.

Almost every water potential question is one of two calculations. The first is solute potential, sometimes called osmotic potential: Ψs = −iCRT. This is the one you use when you are handed a solution and asked how much it pulls on water. The second is total water potential: Ψ = Ψs + Ψp, where Ψp is the pressure potential — the physical squeeze on the water. You need both when you are comparing a cell to the solution around it.

Read the question carefully. If it gives you a molarity and a temperature, it wants −iCRT. If it also gives you a pressure or mentions turgor, it wants the sum. If it gives you Ψs and Ψp directly and asks for Ψ, you only need to add them, and the temperature is a distraction.

• Ψ — water potential, the total. Units of pressure (bars or MPa).
• Ψs — solute potential. Always zero or negative, because dissolving anything lowers water potential.
• Ψp — pressure potential. Positive inside a turgid cell, zero in an open beaker, negative in xylem under tension.
• i — ionization constant (van 't Hoff factor): how many particles one solute unit breaks into. Sucrose and glucose = 1, NaCl and KCl = 2, CaCl₂ and MgCl₂ = 3.
• C — molar concentration of the solution, in mol/L.
• R — the pressure constant. Its value depends on your pressure unit; see the next section.
• T — temperature in kelvin, not Celsius: K = °C + 273.

Bars or megapascals: the mistake that makes you wrong by exactly ten

This is the single most common reason a water potential answer is wrong even though every number was typed in correctly. There are two versions of R in circulation, and they are not interchangeable.

The AP Biology formula sheet works in bars and gives R = 0.0831 L·bar·mol⁻¹·K⁻¹. Plant physiology textbooks, research papers, and most non-AP courses work in megapascals and give R = 0.00831 L·MPa·mol⁻¹·K⁻¹. Those two constants differ by exactly a factor of ten, because 1 MPa = 10 bar. The physics is identical; only the bookkeeping changes.

So if your answer looks like a sensible number but is ten times bigger or smaller than the answer key, you almost certainly used the R that does not match the unit you reported. The calculator above makes you pick the unit first and then prints the R it used right next to the result, so you can never be unsure which convention you are in.

One more trap: if you switch units halfway through a problem, any pressure potential you already wrote down is still in the old unit. A Ψp of 2 bars is 0.2 MPa, not 2 MPa. The calculator will stop and ask you rather than quietly reinterpreting the number.

• Bars → R = 0.0831 L·bar·mol⁻¹·K⁻¹ (AP Biology)
• MPa → R = 0.00831 L·MPa·mol⁻¹·K⁻¹ (plant science, SI)
• 1 MPa = 10 bar. To convert an answer, divide bars by 10 to get MPa.
• Report the unit with every answer. "Ψs = −24.51" is not an answer; "Ψs = −24.51 bars" is.

A worked example: 1.0 M sucrose at 22 °C

This is the classic AP Biology setup, so it is worth doing by hand once. You have a 1.0 molar sucrose solution at room temperature, 22 °C, sitting in an open beaker. Find its water potential in bars.

Step 1, convert the temperature. The gas law needs an absolute temperature, so K = 22 + 273 = 295 K. Leaving 22 in the formula is the second most common error after dropping the minus sign.

Step 2, pick R. You are asked for bars, so R = 0.0831 L·bar·mol⁻¹·K⁻¹.

Step 3, substitute with units attached: Ψs = −(1) × (1.0 mol/L) × (0.0831 L·bar/mol·K) × (295 K).

Step 4, multiply left to right. i × C = 1 × 1.0 = 1.0 mol/L. Then 1.0 × 0.0831 = 0.0831 bar/K. Then 0.0831 × 295 = 24.5145 bar. Apply the minus sign last: Ψs = −24.5145 bars, which you would write as −24.51 bars.

Step 5, check the units cancel: (mol/L) × (L·bar/mol·K) × (K) leaves bar. Moles cancel, litres cancel, kelvin cancels. If you are left with anything else, you used the wrong R or forgot the kelvin conversion.

Step 6, add the pressure potential. The beaker is open to the air, so Ψp = 0, and Ψ = −24.5145 + 0 = −24.51 bars. In megapascals that same solution is −2.45 MPa.

Now put a plant cell into that beaker. Say the cell has a turgor pressure of Ψp = 2 bars and the same internal solute potential. Its water potential is Ψ = −24.5145 + 2 = −22.51 bars. The cell is at −22.51 and the solution outside is at −24.51, so the solution is lower. Water leaves the cell, and it plasmolyses.

Five mistakes that actually cost marks

Every one of these produces a number that looks completely reasonable, which is why they slip through. Check for them before you write your final answer down.

• Dropping the minus sign. Ψs = −iCRT. The negative is part of the formula, not decoration. Solute potential can never be positive.
• Leaving the temperature in Celsius. Using 22 instead of 295 makes your answer roughly thirteen times too small. Convert first, every time.
• Mixing R with the wrong unit. Using R = 0.00831 and then labelling the answer "bars" is wrong by a factor of ten in the direction nobody notices.
• Using i = 1 for a salt. NaCl dissociates into Na⁺ and Cl⁻, so i = 2 and the solution pulls twice as hard as a sucrose solution of the same molarity. CaCl₂ gives three ions, so i = 3.
• Confusing the solution's molarity with the mass of solute. C is mol/L of the whole solution. If the question gives you grams, convert to moles and divide by the volume in litres before you start.
• Bonus: adding Ψs and Ψp with the wrong signs. Ψs is negative and Ψp is usually positive, so the sum is a subtraction in disguise. Write both terms in brackets with their signs showing.

How to check your answer before you hand it in

Three quick tests will catch nearly every arithmetic slip, and none of them takes more than a few seconds.

First, a sign check. Ψs must be zero or negative — never positive. Ψ can be positive only if Ψp is large enough to outweigh Ψs, which happens in a pressurised setup, not in an open beaker. If you got a positive solute potential, you dropped the minus sign.

Second, an order-of-magnitude check. At 22 °C, RT = 0.0831 × 295 ≈ 24.5, so in bars the answer is roughly −24.5 × iC. A 0.3 osmolar solution (say 0.15 M NaCl, where iC = 0.3) should land near −7.4 bars, and it does: −7.35 bars exactly. In MPa the same shortcut is −2.45 × iC. If your answer is nowhere near that, look at your units before you look at your arithmetic.

Third, a unit cross-check. Convert your answer into the other unit and see whether it still looks sensible. −24.51 bars is −2.451 MPa. Plant cells typically sit somewhere between −0.2 and −2 MPa, so a value of −24 MPa should make you suspicious immediately.

One small thing not worth worrying about: whether you use 273 or 273.15 in the kelvin conversion. For the example above that is the difference between −24.5145 and −24.5270 bars — about 0.05%. Use whichever your rubric specifies and do not lose sleep over the last decimal place.