2.3 Dilutions, Alligations & Reconstitution
Key Takeaways
- The fundamental dilution formula C1V1 = C2V2 relies on mass balance, where active solute amount remains constant while volume increases.
- Alligation Alternate calculates exact proportional parts of two different strengths (CH and CL) needed to obtain an intermediate target strength (CT).
- Powder Volume (PV = Vf - Vd) accounts for physical liquid volume displacement by dry powders during reconstitution.
- Displacement Value (DV) represents the volume (in mL) displaced by 1 gram of dry powder.
- Failing to account for displacement values in paediatric oral suspensions or IV reconstitutions causes significant dosing errors.
2.3 Dilutions, Alligations & Reconstitution
Introduction to Compounding & Dilution Mechanics
Pharmaceutical compounding and reconstitution of dry powder formulations require exact mathematical calculations to achieve the intended final drug concentration. In clinical practice across Irish hospital and community pharmacy settings, pharmacists frequently encounter stock solution dilutions, non-standard strength compounding via alligation, and displacement calculations for dry powder oral suspensions and parenteral injections.
1. Dilution Principles & $C_1 V_1 = C_2 V_2$
When diluting a solution by adding diluent (e.g., Water for Injections or 0.9% Sodium Chloride), the total amount of active solute remains constant, while total volume increases and concentration decreases proportionally.
The Fundamental Dilution Formula:
Where:
- $C_1$ = Initial concentration of stock solution
- $V_1$ = Initial volume of stock solution required
- $C_2$ = Final target concentration of diluted solution
- $V_2$ = Final target volume of diluted solution
Diluent Volume Determination:
Note: $C_1$ and $C_2$ must be in identical concentration units (e.g., % w/v, mg/mL, or ratio strength), and $V_1$ and $V_2$ must be in matching volume units (e.g., mL or L).
2. Alligation Alternate & Alligation Medial
When two solutions or semi-solid preparations of different strengths ($C_{\text{High}}$ and $C_{\text{Low}}$) are mixed to obtain an intermediate target strength ($C_{\text{Target}}$), the Alligation Alternate method is used.
Alligation Grid Construction:
Set up the grid with $C_{\text{High}}$ at the top-left, $C_{\text{Low}}$ at the bottom-left, and $C_{\text{Target}}$ in the centre:
C_{\text{High}} & & \text{Parts of } C_{\text{High}} = C_{\text{Target}} - C_{\text{Low}} \\ & C_{\text{Target}} & \\ C_{\text{Low}} & & \text{Parts of } C_{\text{Low}} = C_{\text{High}} - C_{\text{Target}} \end{array}$$ $$\text{Total Parts} = \text{Parts of } C_{\text{High}} + \text{Parts of } C_{\text{Low}} = C_{\text{High}} - C_{\text{Low}}$$ ### Proportional Calculation: $$\text{Quantity of } C_{\text{High}} = \left( \frac{\text{Parts of } C_{\text{High}}}{\text{Total Parts}} \right) \times \text{Total Required Quantity}$$ $$\text{Quantity of } C_{\text{Low}} = \left( \frac{\text{Parts of } C_{\text{Low}}}{\text{Total Parts}} \right) \times \text{Total Required Quantity}$$ --- ## 3. Powder Volume & Displacement Value Calculations Dry powder injections and oral antibiotic suspensions occupy physical volume when reconstituted with liquid diluent. Failing to account for powder displacement leads to significant dosing errors. ### Key Definitions & Formulas: - **Final Reconstituted Volume ($V_f$):** Total volume of liquid after complete reconstitution of powder with diluent. - **Diluent Volume ($V_d$):** Actual volume of liquid added to the dry powder. - **Powder Volume ($PV$):** Volume occupied by the dry powder itself. $$PV = V_f - V_d$$ - **Displacement Value ($DV$):** The volume of liquid (in mL) displaced by a specific mass (typically 1 gram or 1 vial) of dry powder. $$\text{Displacement Value (mL/g)} = \frac{\text{Powder Volume (mL)}}{\text{Mass of Powder (g)}}$$ --- ## 4. Step-by-Step Worked Numerical Examples ### Worked Example 2.3.1: Stock Solution Dilution ($C_1 V_1 = C_2 V_2$) **Clinical Scenario:** A pharmacist needs to prepare 500 mL of a 0.05% w/v Chlorhexidine solution by diluting a 2% w/v Chlorhexidine concentrate stock solution. Calculate: 1. The volume of 2% stock solution required ($V_1$). 2. The volume of purified water diluent to add. **Solution:** - **Step 1: Apply $C_1 V_1 = C_2 V_2$** $$2\% \times V_1 = 0.05\% \times 500\text{ mL}$$ - **Step 2: Solve for $V_1$** $$V_1 = \frac{0.05 \times 500}{2} = \frac{25}{2} = 12.5\text{ mL}$$ - **Step 3: Calculate Diluent Volume** $$\text{Diluent Volume} = V_2 - V_1 = 500\text{ mL} - 12.5\text{ mL} = 487.5\text{ mL}$$ --- ### Worked Example 2.3.2: Alligation Alternate Ointment Compounding **Clinical Scenario:** Prepare 300 g of a 5% w/w Coal Tar ointment by mixing a 20% w/w Coal Tar ointment and a 2% w/w Coal Tar ointment stock. Calculate the exact mass (in grams) of each stock ointment required. **Solution:** - **Step 1: Set Up Alligation Grid** - $C_{\text{High}} = 20\%$ - $C_{\text{Low}} = 2\%$ - $C_{\text{Target}} = 5\%$ - Parts of 20% ointment = $5 - 2 = 3\text{ parts}$ - Parts of 2% ointment = $20 - 5 = 15\text{ parts}$ - Total Parts = $3 + 15 = 18\text{ parts}$ - **Step 2: Calculate Mass of 20% Ointment** $$\text{Mass of 20\%} = \left( \frac{3}{18} \right) \times 300\text{ g} = \frac{1}{6} \times 300\text{ g} = 50\text{ g}$$ - **Step 3: Calculate Mass of 2% Ointment** $$\text{Mass of 2\%} = \left( \frac{15}{18} \right) \times 300\text{ g} = \frac{5}{6} \times 300\text{ g} = 250\text{ g}$$ - **Verification:** $50\text{ g} + 250\text{ g} = 300\text{ g}$. --- ### Worked Example 2.3.3: Antibiotic Reconstitution & Displacement Value **Clinical Scenario:** A vial containing 1 g of Ampicillin dry powder has a known displacement value of 0.7 mL per 1 g vial. 1. If 4.3 mL of Water for Injections is added to the 1 g vial, what is the final reconstituted volume? 2. What is the resulting concentration of Ampicillin in mg/mL? 3. What volume of reconstituted solution must be withdrawn to administer a dose of 250 mg? **Solution:** - **Step 1: Calculate Final Reconstituted Volume ($V_f$)** $$V_f = V_d + PV = 4.3\text{ mL} + 0.7\text{ mL} = 5.0\text{ mL}$$ - **Step 2: Calculate Reconstituted Concentration (mg/mL)** $$\text{Concentration} = \frac{1,000\text{ mg}}{5.0\text{ mL}} = 200\text{ mg/mL}$$ - **Step 3: Calculate Volume for 250 mg Dose** $$\text{Volume} = \frac{250\text{ mg}}{200\text{ mg/mL}} = 1.25\text{ mL}$$ --- ## 5. Method Comparison Table | Technique | Primary Equation / Method | Primary Clinical Application | | :--- | :--- | :--- | | **Simple Dilution** | $C_1 V_1 = C_2 V_2$ | Preparing stock solution dilutions, pediatric IV dilutions | | **Alligation Alternate** | Cross-grid subtraction | Mixing two semi-solid or liquid stock bases to custom % | | **Displacement Reconstitution** | $PV = V_f - V_d$ | Reconstituting IV antibiotic vials and oral suspensions |What volume of a 20% w/v NaCl concentrate stock solution is required to prepare 250 mL of a 0.9% w/v NaCl solution?
Using alligation alternate, calculate the mass of 20% w/w Coal Tar ointment needed to mix with 5% w/w Coal Tar ointment to produce 300 g of 10% w/w Coal Tar ointment.
A 1 g vial of ampicillin dry powder has a displacement volume of 0.7 mL. If 4.3 mL of Water for Injections is added to the vial, what is the final concentration of the reconstituted solution?
When reconstituting an oral antibiotic dry powder suspension, what is the impact on drug concentration if a dispenser ignores powder volume displacement and adds the nominal volume of water directly without checking the final graduation mark?