Osmotic Pressure of Solutions — Theory & Formulas
Osmotic pressure \(\pi\) is the pressure that must be applied to a solution to stop the net flow of
solvent through a semipermeable membrane. It is a colligative property, depending only on the number of
dissolved particles, not their identity.
van ’t Hoff relation (ideal dilute solutions)
For sufficiently dilute solutions, osmotic pressure obeys the gas-law–like relation
\[
\boxed{\ \pi \;=\; i\,C\,R\,T\ }
\]
- \(i\) — van ’t Hoff factor (dimensionless). For a nonelectrolyte, \(i=1\).
For electrolytes, \(i\) approximates the number of solute particles formed per formula unit in solution
(e.g., ideally \(i\approx 2\) for NaCl, \(i\approx 3\) for CaCl\(_2\)), but is often slightly smaller due to ion pairing and non-ideality.
- \(C\) — molar concentration of solute, typically in \(\mathrm{mol\,L^{-1}}\). Internally one may use
\(C\) in \(\mathrm{mol\,m^{-3}}\) with \(R\) in SI units.
- \(R\) — gas constant:
\[
R=8.314462618\ \mathrm{Pa\,m^{3}\,mol^{-1}\,K^{-1}}
\quad\text{or}\quad
R=0.082057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}
\]
Use a unit form consistent with your chosen \(C\) and \(\pi\).
- \(T\) — absolute temperature in kelvin (K). Convert from Celsius with \(T/\mathrm{K}=t/^{\circ}\mathrm{C}+273.15\).
Building the concentration from mass and volume
If the molarity is not supplied directly, compute it from the solute mass \(m\), solute molar mass \(M_r\), and solution volume \(V\):
\[
n \;=\; \frac{m}{M_r}\quad (\mathrm{mol}),\qquad
C \;=\; \frac{n}{V}
\]
with \(m\) in g, \(M_r\) in \(\mathrm{g\,mol^{-1}}\), and \(V\) either in L (to get \(C\) in \(\mathrm{mol\,L^{-1}}\)) or in \(\mathrm{m^{3}}\) (to get \(C\) in \(\mathrm{mol\,m^{-3}}\)).
Rearrangements (solving for other variables)
The van ’t Hoff relation easily inverts to determine unknown quantities when the others are known:
\[
\begin{aligned}
\textbf{Molar mass } M_r &: \quad
M_r \;=\; \frac{i\,m\,R\,T}{\pi\,V}
\quad\big(\text{osmometry; use } m\text{ in g, }V\text{ in }\mathrm{m^{3}}\text{ with }R \text{ in SI}\big) \\
\textbf{Molarity } C &: \quad
C \;=\; \frac{\pi}{i\,R\,T} \\
\textbf{Mass } m &: \quad
m \;=\; \frac{\pi\,V\,M_r}{i\,R\,T} \\
\textbf{Volume } V &: \quad
V \;=\; \frac{i\,m\,R\,T}{\pi\,M_r} \\
\textbf{Temperature } T &: \quad
T \;=\; \frac{\pi}{i\,R\,C} \\
\textbf{van ’t Hoff factor } i &: \quad
i \;=\; \frac{\pi}{C\,R\,T}
\end{aligned}
\]
Units & consistency tips
- Always use kelvin for \(T\).
- If you choose \(\pi\) in \(\mathrm{atm}\) and \(C\) in \(\mathrm{mol\,L^{-1}}\), use \(R=0.082057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\).
- If you choose \(\pi\) in \(\mathrm{Pa}\) and \(C\) in \(\mathrm{mol\,m^{-3}}\), use \(R=8.314462618\ \mathrm{Pa\,m^{3}\,mol^{-1}\,K^{-1}}\).
- The calculator supports many pressure units (Pa, kPa, MPa, bar, mbar, atm, mmHg, Torr, psi) and converts them consistently.
Worked examples
1) Nonelectrolyte (glucose) — compute \(\pi\)
\(m=1.00\ \mathrm{g}\) glucose, \(M_r=180.156\ \mathrm{g\,mol^{-1}}\), \(V=0.100\ \mathrm{L}\) at \(25.0^{\circ}\mathrm{C}\) (\(T=298.15\ \mathrm{K}\)), \(i=1\).
\[
\begin{aligned}
n &= \frac{1.00\ \mathrm{g}}{180.156\ \mathrm{g\,mol^{-1}}} = 5.55\times 10^{-3}\ \mathrm{mol} \\
C &= \frac{n}{V} = \frac{5.55\times 10^{-3}\ \mathrm{mol}}{0.100\ \mathrm{L}} = 5.55\times 10^{-2}\ \mathrm{mol\,L^{-1}} \\
\pi &= i\,C\,R\,T
= (1)\,(5.55\times 10^{-2})\ \mathrm{mol\,L^{-1}}\,(0.082057)\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\,(298.15\ \mathrm{K}) \\
&\approx 1.36\ \mathrm{atm}
\end{aligned}
\]
2) Electrolyte (NaCl) — non-ideal \(i\)
Take \(C=0.100\ \mathrm{mol\,L^{-1}}\), \(T=298.15\ \mathrm{K}\), and an effective \(i=1.8\) (to reflect partial ion pairing).
\[
\pi = i\,C\,R\,T
= (1.8)\,(0.100)\ \mathrm{mol\,L^{-1}}\,(0.082057)\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\,(298.15\ \mathrm{K})
\approx 4.40\ \mathrm{atm}
\]
Assumptions & limitations
- Dilute solutions. The linear van ’t Hoff law holds best at low \(C\). At higher concentrations, deviations appear. A common refinement is the osmotic virial expansion:
\(\displaystyle \frac{\pi}{C} = \frac{R\,T}{M_r} + B\,C + C'\,C^{2} + \cdots\) (important for macromolecules).
- Ideal semipermeable membrane. Real membranes may allow some solute passage or retain bound solvent; measured \(\pi\) can differ slightly.
- Effective particle count. The van ’t Hoff factor \(i\) is an apparent value reflecting dissociation/association and activity effects at the working concentration and temperature.
- Nonvolatile solute. The treatment assumes solute does not exert significant vapor pressure and that volume change upon mixing is negligible (reasonable for dilute solutions).
How the calculator uses this theory
- You may provide \(C\) directly or have the tool compute it from \(m\), \(M_r\), and \(V\).
- For electrolytes, set \(i\). If unknown, the tool can solve for \(i\) from a measured \(\pi\).
- The solver can rearrange the relation to find \(M_r\), \(m\), \(V\), \(T\), or \(C\) when \(\pi\) is known (use consistent units).
- Calculation steps show unit conversions explicitly (e.g., between Pa and atm) and present multi-line derivations for transparency.
