Mixture Problems in Pharmaceutical Sciences

Abstract

In this chapter, I have approached general mixture composition problems as regular mathematical mixture problems. Equations are developed to represent the different rates and quantities of the variables that are involved in the mixture problems. These equations are used to identify the domain of all variables and develop a procedure to provide all possible answers. You will learn to identify the type of variables that are preserved during the initial and final states of a process, construct equations that represent the initial and final states of these variables, and solve the system of linear equations to determine the specific values of these variables. You may skip the three- and four-component mixture composition problems if you feel that you will not be needing those and move to the empirical methods of alligation. The obscured method of alligation alternate is unraveled for the first time. The rules of the method and the advantages and disadvantages of it are clearly explained, and their application is herein expanded to other areas such as physical chemistry, pharmacokinetics, and pharmacoeconomics.

Appendices

Exercises

• 6.1. The surface area of cholesterol is 35 Å²/molecule, whereas that of dioleoylphosphocholine (DOPE) is 55 Å²/molecule. At what molar ratio and amounts do you have to mix a total of 5.2 ∙ 10¹⁸ molecules to achieve a molecular area of 44.4 Å²/molecule assuming an ideal interaction? (Avogadro number, A_N = 6.022 ∙ 10²³ molecules/mol, MW (cholesterol) = 386.65 g/mol, MW (DOPE) = 744 g/mol)

  • (a) Identify the variables that are combined in this problem, (b) construct equations for the conservation of those variables in the initial and final states of the system, and (c) solve the system of equations to calculate the values of the variables

    • (Answer: mole ratio n₁/n₂ = 1.13, m₁ = 1.77 mg, m₂ = 3.02 mg)

• 6.2. Experiments were carried out on a Langmuir apparatus at mole fraction x_Chol of 0.9 to determine the molecular dimensions of cholesterol and DOPE as described in Example 6.1. The measured dimensions of the molecular complex were found to be 33 Å²/molecule. No information is given about the number of total molecules used in the experiments. Find the deviation from ideality and explain your results. (A₁ = 35 Å²/molecule, A₂ = 55 Å²/molecule, experimental A_f,exp = 33 Å²/molecule, Avogadro number, A_N = 6.022 ∙ 10²³ molecules/mol, MW (cholesterol) = 386.65 g/mol, MW (DOPE) = 744 g/mol, 1 Angstrom, Å = 10⁻¹⁰ m)

  • (Answer: = −4 Å²/molecule)

• 6.3. Use the Raoult’s law to calculate the molar ratio and amounts that you have to use to mix a total of 10 mmoles of sevoflurane (general anesthetic) and water at 20 °C to achieve a vapor pressure of 22.1 mmHg over the solution? (A_N = 6.022 ∙ 10²³ molecules/mol, MW (C₄H₃F₇O) = 200 g/mol, MW (H₂O) = 18 g/mol, ${P}_{\mathrm{sevofluran}}^{\ast }={P}_1^{\ast }=157\;\mathrm{mmHg}$, ${P}_{\mathrm{water}}^{\ast }={P}_2^{\ast }=17.5\;\mathrm{mmHg}$ at 20 °C)

  Identify the variables that are combined in this problem, construct equations for the conservation of those variables in the initial and final states of the system, and solve the equations to calculate the values of the variables

• (Answer: n₂/n₁ = 29.3, n₁ = 0.33 mmol, n₂ = 9.67 mmol, m₁ = 66 mg, m₂ = 174.1 mg)

• 6.4. The prices for two multivitamin brands are:

  • Brand 1 (R₁): 6.25 cents a tablet

  • Brand 2 (R₂): 3.75 cents a tablet

    If the cost has to be maintained at 4.5 cents per tablet and the available budget (C_f) is $450, how much of each brand should be purchased? (a) Identify the two variables that are combined in this problem, (b) construct equations for the conservation of those variables for the initial and final states of the system, and (c) solve the system of those equations to calculate the values of the variables.

• (Answer: T₁ = 3000 tablets, $187.5; T₂ = 7000 tablets, $262.5)

• 6.5. Two different types of sustained drug release granules release a drug at a rate of 21.3 and 3.4 mg/h. Calculate the weight ratio of the two granules if 1.8 g of total mixture released the drug at an average rate of 12 mg/h.

  • (Answer: M₁ = 864.8 mg, M₂ = 935.2 mg, $\frac{M_1}{M_2}=0.92$)

• 6.6. At what quantity should you mix lidocaine with 60 g of white petrolatum to make an ointment that contains 2%w/w lidocaine?

  • (Answer: 1.224 g)

• 6.7. You mix 25 g of 0.5%w/w tetracaine cream with 23 g of 1% tetracaine cream. What is the concentration of tetracaine in the mixture?

  • (Answer: 0.74%w/w)

• 6.8. At what ratio do you have to mix 8%w/v drug solution with a 5%w/v drug solution to prepare 1 L of 2%w/v drug solution?

• 6.9. You plan to mix 70% H₂SO₄ (specific gravity = 1.65) with 5%w/v H₂SO₄ to make 1650 mL of 30%w/w sulfuric acid (specific gravity = 1.24). How much of each should you use?

  • (Answer: V₁ = 480.8 mL; V₂ = 1169.2 mL)

• 6.10. Alcohol USP (253 mL) was mixed with 0.464 L of 27%v/v alcohol in water. What is the concentration of water in the mixture? (Assume that volumes are additive)

  • (Answer: 49% v/v)

• 6.11. How many milligrams of vecuronium (short-acting muscle relaxant) should you add to 0.783 g of NaCl in order to prepare a 1%w/w powder mixture?

  • (Answer: 7.91 mg)

• 6.12. What is the concentration of sucrose in a mixture of 250 mL of 80%w/w sucrose (ρ = 1.45) with 620 mL of 7%w/v sucrose solution (ρ = 1.03)?

  • (Answer: 38.32% w/v)

• 6.13. (a) How many milliliters of 80%w/v chloroquine phosphate must be added to 0.750 L of 45%w/v chloroquine phosphate to make a solution containing 63.5% chloroquine phosphate?

  (b) If chloroquine phosphate is 60% chloroquine base, what is the concentration of chloroquine base in the final mixture?

  • (Answer: (a) 840.91 mL (b) 38.1% w/v)

• 6.14. A patient is infused with metronidazole 5%w/v. You receive a request from the physician to double-concentrate the solution. You have on hand metronidazole solutions 150 and 5 mg/mL. How much of each should you use to prepare 4 L of the double-concentrate metronidazole solution?

  • (Answer: V₁ = 2.62 L; V₂ = 1.38 L)

• 6.15. What is the percentage w/w of magnesium chloride (MgCl₂) in a powder mixture containing: 1.5 g of 0.1% MgCl₂ in dextrose, 200 mg of 0.45% MgCl₂ in dextrose, and 0.35 g of polyethylene glycol?

  • (Answer: 0.117% w/w)

• 6.16. In preparing IV bags of (80 mg trimethoprim and 400 mg sulfamethoxazole)/5 mL, you must mix (1 mg trimethoprim and 5 mg sulfamethoxazole)/mL solution with (0.4 g trimethoprim and 2 g sulfamethoxazole)/5 mL solution (each solution contains both drugs at the given concentrations). How much of each should you use to prepare four 1 L IV bags for infusion?

  • (Answer: V₁ = 3.24 L; V₂ = 0.76 L)

• 6.17. How many microliters of oil and how many milliliters of 2.5%w/v dihydrotachysterol (DHT) in oil are needed to make 4 mL of 20 mg/mL DHT solution in oil?

  • (Answer: V₁ = 800 μL; V₂ = 3.2 mL)

• 6.18. Given the following formula:

  Lugol’s solution (5%w/v iodine, 10%w/v KI in water)

  Iodine tincture USP (2%w/v iodine, 2.4%w/v NaI in 47%v/v ethanol)…

  M. et ft. 2 fξ of 3.8% iodine tincture

  • (a) Calculate the volume of each solution needed to prepare the formula.

  • (b) How much ethanol is in the final product?

  • (c) What is the iodide concentration in the final solution?

    • (Answer: (a) V₁ = 35.48 mL, V₂ = 23.66 mL; (b) 11.12 mL; (c) 6.96%)

Additional Exercises

• 6.19. You have on hand four 4 mL vials of suramin 0.01 g/mL solution and eight 4 mL vials of suramin 0.5%w/v solution.

  • (a) How much of a 0.65%w/v suramin solution can you make by mixing the two solutions?

  • (b) How much of a 0.65% suramin solution can you make by first mixing the two solutions and then diluting the remaining of the concentrated solution with sterile water?

  • (c) Use the method of alligation alternate to calculate the volume of a 0.65% suramin solution that you can make from the two solutions and sterile water (0%w/v)? Compare your answer with (a).

    • (Answer: (a) 45.7 mL; (b) 49.24 mL; (c) 30 mL)

• 6.20. The nutritive ratio of a dietary supplement is defined as the ratio of calories obtained from proteins to the calories obtained from carbohydrates and fat per gram of food supplement. You have obtained bulk powder of two dietary supplements with nutritive ratio R₁ = (2.3/1)/g and R₂ = (6.7/1)/g. How many grams of each supplement should you use to prepare 2.2 lb of a protein supplement having a nutritive ratio of (4/1)/g?

  (a) Identify the two variables that are combined in this problem, (b) construct equations for the conservation of those variables for the initial and final states of the system, and (c) solve the system of those equations to calculate the values of the variables.

  • (Answer: M₁ = 612.9 g, M₂ = 385.6 g)

• 6.21. You have developed three different transdermal patches of drug D. The first one releases the drug in the blood of a patient and achieves a concentration of 0.34 mg/dL/h. The second patch releases the drug at a rate of 0.42 mg/dL/h and the third one at a rate of 0.04 mg/dL/h.

  • (a) In what way could these three patches be applied on patient’s skin (sequentially) in order for the drug to reach therapeutic levels of 0.25 mg of drug per deciliter of blood, within exactly 1 h? Identify the time domain for the three patches (how long could each patch be left on the skin if the total application time is 60 min).

  • (b) If the third patch (0.04 mg/dL/h) is applied first and is left for exactly 20 min on the skin of the patient, how long should you apply the other two patches in order to achieve a concentration of 0.25 mg/dL within exactly 60 min from the application of the third patch?

    • (Answer: (a) 0 < t₁ < 42, 0 < t₂ < 33.16, and 18 < t₃ < 26.84; (b) t₁ = 32.5 min, t₂ = 7.5 min)

• 6.22. Urine production per unit time from three perfused rat kidneys is tabulated in the table below:

  	Rate of urine accumulation (mL/min)
  Kidney #1	1
  Kidney #2	0.8
  Kidney #3	0.5

  Assuming that you collected 2.8 mL of urine within 4 min, (a) what is the time domain for the duration of urine collection and (b) what is the urine volume collected from kidney #3 at minimum and maximum urine collection times?

• (Answer: (a) 0 < t₁ < 1.6, 0 < t₂ < 2.67, and 1.33 < t₃ < 2.4 min; (b) 0.667, 1.2 mL)

• 6.23. In Exercise 6.22, for how long do you have to collect urine from kidney #3 if you collected urine from the other two kidneys at equal time?

  • (Answer: t₁ = t₂ = 1 min; t₃ = 2 min)

• 6.24. Solve Exercise 6.22 using the method of alligation alternate.

  • (Answer: t₁ = t₂ = 1 min; t₃ = 2 min)

• 6.25. The nutritive ratio of a dietary supplement is defined as the ratio of calories obtained from proteins to the calories obtained from carbohydrates and fat per gram of food. Without defining the domain of each supplement, determine at least three combinations and their gram amounts at which, when they combine, they make 1.5 kg of a supplement having a nutritive ratio of 1/15. The dietary supplements are given below in the table.

  Dietary supplement	Nutritive ratio
  #1	1/8.5
  #2	1/9.5
  #3	1/4.25
  #4	1/38
  #5	1/32

• (Answer: Combination 1: M₁ = M₂ = M₃ and M₄ = M₅; Combination 2: M₁ = M₂ = M₃ combined with M₄ to make M_f/2 of R_f and M₁ = M₂ = M₃ is combined with M₅ to make the remaining M_f/2 of R_f. These are two different mixtures that make the same R_f; Combination 3: M₁ = M₂ combined with M₄ to make M_f/2 of R_f and M₃ is combined with M₅ to make the remaining M_f/2 of R_f. These are two different mixtures that make the same R_f.)

• 6.26. The price for four multivitamin brands are:

  Brand 1 (R₁): 4.50 cents a tablet

  Brand 2 (R₂): 6.99 cents a tablet

  Brand 3 (R₃): 3.99 cents a tablet

  Brand 4 (R₄): 2.45 cents a tablet

  If the cost has to be maintained at 3.5 cents per tablet and the available budget (B) is $3150, how much of each brand should be purchased?

  (Answer: Domain of brand 4 {T₄|T₄ (28,636, 69,185) tablets})