Q. how to calculate osmotic pressure

Step 1. What osmotic pressure is, conceptually. Osmotic pressure is the extra hydrostatic pressure that must be applied to the solution side of a semipermeable membrane to stop the net flow of solvent into the solution. This is a colligative property, which means it depends on the number of solute particles per unit volume rather than their chemical identity.

Step 2. The basic equation for dilute solutions (van ‘t Hoff law). For an ideal dilute solution the osmotic pressure \( \Pi \) behaves like the pressure of an ideal gas of solute particles. The fundamental form is

\[ \Pi V = n R T \]

where \( \Pi \) is the osmotic pressure, \( V \) is the solution volume, \( n \) is the number of moles of solute present in volume \( V \), \( R \) is the universal gas constant, and \( T \) is the absolute temperature in kelvin.

Step 3. Expressing the law in terms of molarity. Molarity \( M \) is defined as moles of solute per liter, so \( n = M V \) when \( V \) is expressed in liters. Substitute \( n = M V \) into the previous equation and cancel \( V \) to obtain the commonly used form:

\[ \Pi = M R T \]

Step 4. Accounting for electrolytes (van ‘t Hoff factor). If the solute dissociates into multiple particles in solution, include the van ‘t Hoff factor \( i \), which is the average number of particles produced per formula unit of solute. The practical formula is

\[ \Pi = i M R T \]

Here \( i \) equals 1 for a nonelectrolyte that does not dissociate, 2 for an ideal fully dissociated NaCl, 3 for an ideal fully dissociated CaCl2, and so on. For real solutions \( i \) can differ from the integer ideal value because of incomplete dissociation and ionic interactions.

Step 5. Units and the gas constant. Use consistent units. If \( M \) is in moles per liter and you want \( \Pi \) in atmospheres, use \( R = 0.082057 \) liter·atm·mol^{-1}·K^{-1}. To obtain \( \Pi \) in pascals or kilopascals, use \( R = 8.3144621 \) J·mol^{-1}·K^{-1} and express volume in cubic meters so that pressure comes out in pascals. Remember that 1 atm = 101325 Pa = 101.325 kPa.

Step 6. Worked numerical example. Calculate the osmotic pressure of an ideal 0.10 mol·L^{-1} sodium chloride solution at 25 degrees Celsius assuming full dissociation.

Step 6a. Identify variables. Molarity \( M = 0.10 \) mol·L^{-1}. Van ‘t Hoff factor \( i = 2 \) for full dissociation of NaCl. Temperature \( T = 25 + 273.15 = 298.15 \) K. Gas constant \( R = 0.082057 \) L·atm·mol^{-1}·K^{-1} if we want pressure in atmospheres.

Step 6b. Substitute into the formula \( \Pi = i M R T \) and compute. First compute the product \( R T \):

\[ R T = 0.082057 \times 298.15 \approx 24.47 \ \text{L·atm·mol}^{-1} \]

Then multiply by \( i M \):

\[ \Pi = i M R T = 2 \times 0.10 \times 24.47 \approx 4.89 \ \text{atm} \]

Step 6c. Convert to kilopascals if desired. Use 1 atm = 101.325 kPa.

\[ \Pi \approx 4.89 \times 101.325 \approx 495 \ \text{kPa} \]

Step 7. Notes about nonideal behavior and other exact thermodynamic forms. For concentrated or nonideal solutions ionic interactions and activity effects become important. A more general thermodynamic expression relates osmotic pressure to solvent activity or chemical potential. Two useful forms are

\[ \Pi = -\dfrac{R T}{\bar V_1} \ln a_1 \]

and, for an incompressible solvent and expressed via mole fraction \( x_1 \) approximately,

\[ \Pi \approx -\dfrac{R T}{\bar V_1} \ln x_1 \]

In these formulas \( \bar V_1 \) is the partial molar volume of the solvent, and \( a_1 \) is the solvent activity. For dilute solutions the logarithm can be linearized and these expressions reduce to the van ‘t Hoff form shown above.

Step 8. Summary checklist to calculate osmotic pressure in practice.

1. Decide whether the solution is dilute and ideal enough to use the van ‘t Hoff law. 2. Determine molarity \( M \), temperature \( T \) in kelvin, and the appropriate van ‘t Hoff factor \( i \). 3. Choose \( R \) consistent with desired pressure units. 4. Compute \( \Pi = i M R T \). 5. For nonideal or concentrated solutions, use activity or osmotic coefficient data and the thermodynamic expressions given above.