Pharmaceutical Production & Formulation
Differential, Mathematical & Case Study Interview Questions
Advanced Technical Q&A for Pharma Manufacturing Professionals
1. Derive the Noyes-Whitney equation and explain each term.
dC/dt = k * A * (Cs - C)
Where:• dC/dt = dissolution rate
• k = dissolution rate constant = D/h (D = diffusion coefficient, h = diffusion layer thickness)
• A = surface area of particle
• Cs = saturation solubility
• C = concentration in bulk solution
This equation forms the basis for understanding how particle size reduction increases dissolution rate.
2. Calculate the Reynolds number for fluid flow in a mixing tank with diameter 0.5m, impeller speed 200 rpm, fluid density 1050 kg/m³, viscosity 0.1 Pa·s.
Re = (ρ * N * D²) / μ
N = 200 rpm = 200/60 = 3.33 rps
Re = (1050 * 3.33 * 0.5²) / 0.1
Re = (1050 * 3.33 * 0.25) / 0.1
Re = 873.825 / 0.1 = 8738.25
Since Re > 4000, the flow is turbulent.
N = 200 rpm = 200/60 = 3.33 rps
Re = (1050 * 3.33 * 0.5²) / 0.1
Re = (1050 * 3.33 * 0.25) / 0.1
Re = 873.825 / 0.1 = 8738.25
3. Solve the first-order degradation kinetics equation for a drug with k = 0.015 day⁻¹. Calculate shelf life (t90).
ln(C/C₀) = -kt
For t₉₀: C/C₀ = 0.9
ln(0.9) = -0.015 * t₉₀
t₉₀ = -ln(0.9)/0.015
t₉₀ = 0.10536/0.015 = 7.024 days
Shelf life (90% potency) = 7 days.
For t₉₀: C/C₀ = 0.9
ln(0.9) = -0.015 * t₉₀
t₉₀ = -ln(0.9)/0.015
t₉₀ = 0.10536/0.015 = 7.024 days
4. Derive the equation for Fick's first law of diffusion in pharmaceutical context.
J = -D * (dC/dx)
Where:• J = diffusion flux (mol/m²·s)
• D = diffusion coefficient (m²/s)
• dC/dx = concentration gradient
The negative sign indicates diffusion occurs from high to low concentration. This governs drug release from matrices and membrane transport.
5. Calculate the specific surface area of spherical particles with diameter 50 μm and density 1.2 g/cm³.
Surface area of sphere = 4πr²
Volume = (4/3)πr³
r = 25 μm = 25 × 10⁻⁶ m
SA = 4π(25×10⁻⁶)² = 7.854×10⁻⁹ m² per particle
Volume = (4/3)π(25×10⁻⁶)³ = 6.545×10⁻¹⁴ m³ per particle
Mass = Volume × Density = 6.545×10⁻¹⁴ × 1200 = 7.854×10⁻¹¹ kg
Specific surface area = SA/mass = 7.854×10⁻⁹ / 7.854×10⁻¹¹ = 100 m²/kg
Volume = (4/3)πr³
r = 25 μm = 25 × 10⁻⁶ m
SA = 4π(25×10⁻⁶)² = 7.854×10⁻⁹ m² per particle
Volume = (4/3)π(25×10⁻⁶)³ = 6.545×10⁻¹⁴ m³ per particle
Mass = Volume × Density = 6.545×10⁻¹⁴ × 1200 = 7.854×10⁻¹¹ kg
Specific surface area = SA/mass = 7.854×10⁻⁹ / 7.854×10⁻¹¹ = 100 m²/kg
6. Solve the power number equation for mixing: P = Nₚ * ρ * N³ * D⁵. Calculate power for Nₚ=5, ρ=1000 kg/m³, N=4 rps, D=0.3m.
P = 5 * 1000 * 4³ * 0.3⁵
P = 5 * 1000 * 64 * 0.00243
P = 5 * 1000 * 0.15552
P = 777.6 W = 0.78 kW
P = 5 * 1000 * 64 * 0.00243
P = 5 * 1000 * 0.15552
P = 777.6 W = 0.78 kW
7. Derive the equation for Stokes' law sedimentation velocity.
v = (2r²(ρₚ - ρₗ)g) / (9η)
Where:
• v = terminal settling velocity
• r = particle radius
• ρₚ = particle density
• ρₗ = liquid density
• g = gravitational acceleration
• η = liquid viscosity
Critical for suspension formulation and particle size analysis.
Where:
• v = terminal settling velocity
• r = particle radius
• ρₚ = particle density
• ρₗ = liquid density
• g = gravitational acceleration
• η = liquid viscosity
Critical for suspension formulation and particle size analysis.
8. Calculate the Arrhenius equation parameters: k₂/k₁ = exp[(Ea/R)(1/T₁ - 1/T₂)]. If k doubles from 25°C to 35°C, find Ea.
T₁ = 298K, T₂ = 308K, k₂/k₁ = 2
ln(2) = (Ea/R)(1/298 - 1/308)
0.6931 = (Ea/8.314)(0.003356 - 0.003247)
0.6931 = (Ea/8.314)(0.000109)
Ea = (0.6931 * 8.314) / 0.000109
Ea ≈ 52,850 J/mol = 52.85 kJ/mol
ln(2) = (Ea/R)(1/298 - 1/308)
0.6931 = (Ea/8.314)(0.003356 - 0.003247)
0.6931 = (Ea/8.314)(0.000109)
Ea = (0.6931 * 8.314) / 0.000109
Ea ≈ 52,850 J/mol = 52.85 kJ/mol
9. Solve for tablet tensile strength: σ = 2F/(πDt), where F=1000N, D=10mm, t=3mm.
σ = (2 * 1000) / (π * 0.01 * 0.003)
σ = 2000 / (9.4248×10⁻⁵)
σ = 21.22 × 10⁶ Pa = 21.22 MPa
σ = 2000 / (9.4248×10⁻⁵)
σ = 21.22 × 10⁶ Pa = 21.22 MPa
10. Derive the equation for moisture diffusion during drying using Fick's second law.
∂C/∂t = D * (∂²C/∂x²)
Where C is moisture concentration, t is time, x is distance, D is moisture diffusivity. This partial differential equation describes falling rate period in drying.
11. Case Study: A tablet formulation shows capping during compression. Provide differential diagnosis and solution strategy.
Possible Causes:
1. Excessive elastic recovery (high viscoelasticity)
2. Air entrapment
3. Improper granulation moisture
4. Incorrect machine settings
Investigation Protocol:
• Measure tablet tensile strength and elasticity
• Check moisture content (LOD 1-3% optimal)
• Evaluate granulation flow properties
• Analyze compression force profile
Solutions:
• Add plastic binder (MCC, HPMC)
• Use pre-compression
• Optimize moisture content
• Reduce compression speed
1. Excessive elastic recovery (high viscoelasticity)
2. Air entrapment
3. Improper granulation moisture
4. Incorrect machine settings
Investigation Protocol:
• Measure tablet tensile strength and elasticity
• Check moisture content (LOD 1-3% optimal)
• Evaluate granulation flow properties
• Analyze compression force profile
Solutions:
• Add plastic binder (MCC, HPMC)
• Use pre-compression
• Optimize moisture content
• Reduce compression speed
12. Mathematical Modeling: A sustained-release matrix tablet follows Higuchi kinetics. Derive the equation and calculate release at 4 hours if k=0.15 mg/√h.
Q = k√t (Higuchi equation for matrix systems)
Q = 0.15 * √4 = 0.15 * 2 = 0.30 mg released per cm²
This square root time dependence indicates diffusion-controlled release from insoluble matrix.
Q = 0.15 * √4 = 0.15 * 2 = 0.30 mg released per cm²
13. Case Study: An emulsion shows creaming after 1 week. Perform root cause analysis with mathematical verification.
Stokes' Law Analysis:
v = (2r²Δρg)/(9η)
To reduce creaming rate:
1. Reduce droplet size (r² term dominant)
2. Match density (Δρ → 0)
3. Increase viscosity (η)
Corrective Actions:
• Homogenize to reduce droplet size < 1μm
• Add viscosity modifier (xanthan gum)
• Use density matching oils
• Optimize HLB of emulsifier
v = (2r²Δρg)/(9η)
To reduce creaming rate:
1. Reduce droplet size (r² term dominant)
2. Match density (Δρ → 0)
3. Increase viscosity (η)
Corrective Actions:
• Homogenize to reduce droplet size < 1μm
• Add viscosity modifier (xanthan gum)
• Use density matching oils
• Optimize HLB of emulsifier
14. Solve: If a powder blend has Carr's Index = 25 and Hausner Ratio = 1.33, classify flow properties and suggest improvements.
Carr's Index = [(Tapped - Bulk)/Tapped] × 100
Hausner Ratio = Tapped/Bulk
Classification: Poor flow (Carr's 20-35, Hausner 1.25-1.5)
Required: CI < 15, HR < 1.25 for good flow
Solutions: Add glidant (0.5-2% colloidal silica), optimize particle size distribution, granulate.
Hausner Ratio = Tapped/Bulk
Classification: Poor flow (Carr's 20-35, Hausner 1.25-1.5)
Required: CI < 15, HR < 1.25 for good flow
15. Case Study: A lyophilized product shows collapse during freeze-drying. Provide thermal analysis solution.
Collapse Temperature (Tc) Analysis:
• Measure by DSC/FDM
• Typically -20°C to -35°C for sugars
Critical Process Parameters:
1. Shelf temperature must remain < Tc during primary drying
2. Chamber pressure control (0.1-0.2 mBar)
3. Ramp rate optimization
Formulation Solutions:
• Add collapse protectants (trehalose, dextran)
• Optimize solid content (5-20%)
• Adjust buffer concentration
• Measure by DSC/FDM
• Typically -20°C to -35°C for sugars
Critical Process Parameters:
1. Shelf temperature must remain < Tc during primary drying
2. Chamber pressure control (0.1-0.2 mBar)
3. Ramp rate optimization
Formulation Solutions:
• Add collapse protectants (trehalose, dextran)
• Optimize solid content (5-20%)
• Adjust buffer concentration
16. Calculate mixing time using dimensional analysis: θ = k * (D²/ν)^a * (ν/ND²)^b
Where: θ = mixing time, D = impeller diameter, ν = kinematic viscosity, N = rotational speed
For turbulent flow: θ ∝ 1/N
For laminar flow: θ ∝ μ/ρND²
Experimental determination needed for specific system.
For turbulent flow: θ ∝ 1/N
For laminar flow: θ ∝ μ/ρND²
Experimental determination needed for specific system.
17. Case Study: Content uniformity failure in low-dose tablets (0.1 mg drug). Provide statistical and formulation solution.
Statistical Analysis:
According to binomial distribution:
σ = √[np(1-p)] where n = particles per tablet
To achieve RSD < 5%: Need n > 400 drug particles per tablet
Solution Strategy:
1. Micronize drug to increase particle number
2. Use geometric dilution technique
3. Prepare ordered mix (drug adsorbed on carrier)
4. Implement in-process weight control ±2%
According to binomial distribution:
σ = √[np(1-p)] where n = particles per tablet
To achieve RSD < 5%: Need n > 400 drug particles per tablet
Solution Strategy:
1. Micronize drug to increase particle number
2. Use geometric dilution technique
3. Prepare ordered mix (drug adsorbed on carrier)
4. Implement in-process weight control ±2%
18. Derive the Michaelis-Menten equation for enzyme degradation in liquid formulations.
v = (V_max * [S]) / (K_m + [S])
Where:
• v = degradation rate
• V_max = maximum rate
• [S] = substrate concentration
• K_m = Michaelis constant
Important for predicting protease degradation in protein formulations.
Where:
• v = degradation rate
• V_max = maximum rate
• [S] = substrate concentration
• K_m = Michaelis constant
Important for predicting protease degradation in protein formulations.
19. Case Study: Tablet dissolution fails USP specification at 45 minutes but passes at 30 minutes. Provide root cause investigation.
Possible Causes:
1. Gel layer formation (hydrophilic polymers)
2. Poor disintegration (hard tablets)
3. Drug polymorphism change
4. Excipient interaction
Investigation Plan:
• Conduct dissolution profile (5, 10, 15, 30, 45, 60 min)
• Analyze disintegration time vs dissolution correlation
• Perform XRD for polymorph detection
• Check storage conditions (moisture uptake)
1. Gel layer formation (hydrophilic polymers)
2. Poor disintegration (hard tablets)
3. Drug polymorphism change
4. Excipient interaction
Investigation Plan:
• Conduct dissolution profile (5, 10, 15, 30, 45, 60 min)
• Analyze disintegration time vs dissolution correlation
• Perform XRD for polymorph detection
• Check storage conditions (moisture uptake)
20. Calculate the D-value for sterilization: D = t/(log N₀ - log N). If N₀=10⁶, N=10⁻³ after 15 minutes, find D.
D = 15 / (log₁₀10⁶ - log₁₀10⁻³)
D = 15 / (6 - (-3)) = 15/9 = 1.67 minutes
This means 90% reduction in 1.67 min at given temperature.
D = 15 / (6 - (-3)) = 15/9 = 1.67 minutes
This means 90% reduction in 1.67 min at given temperature.
21. Solve population balance equation for milling: ∂n/∂t + ∂(Gn)/∂L = B - D
Where: n = number density function, G = growth rate, L = particle size, B = birth rate, D = death rate.
This integro-differential equation describes particle size evolution during milling.
This integro-differential equation describes particle size evolution during milling.
22. Case Study: Hot melt extrusion shows drug degradation at processing temperature. Provide thermal analysis solution.
Thermal Analysis Required:
1. DSC/TGA to determine melting point and degradation temperature
2. Rheology measurement at processing temperature
Mathematical Modeling:
Use Arrhenius to predict degradation: k = A exp(-Ea/RT)
For 10°C reduction (383K to 373K):
k₂/k₁ = exp[(Ea/R)(1/373 - 1/383)]
With Ea=100 kJ/mol: k reduces by ~50%
1. DSC/TGA to determine melting point and degradation temperature
2. Rheology measurement at processing temperature
Mathematical Modeling:
Use Arrhenius to predict degradation: k = A exp(-Ea/RT)
For 10°C reduction (383K to 373K):
k₂/k₁ = exp[(Ea/R)(1/373 - 1/383)]
With Ea=100 kJ/mol: k reduces by ~50%
23. Calculate the power required for agitation of non-Newtonian fluid using Metzner-Otto concept.
For power-law fluid: τ = K(γ̇)^n
Effective shear rate: γ̇_eff = k_s * N
Where k_s = 10 for turbines
Apparent viscosity: η_app = K(γ̇_eff)^(n-1)
Re_app = (ρND²)/η_app
Then use power number vs Re curve.
Effective shear rate: γ̇_eff = k_s * N
Where k_s = 10 for turbines
Apparent viscosity: η_app = K(γ̇_eff)^(n-1)
Re_app = (ρND²)/η_app
Then use power number vs Re curve.
24. Case Study: Spray drying yields high fines (>40%). Provide mathematical optimization.
Droplet Size Analysis:
d₃₂ = (σ/ρ)^0.6 * (μ/σ)^0.1 * (Q/ΔP)^0.4 (for pressure nozzle)
Where d₃₂ = Sauter mean diameter
Optimization Parameters:
1. Increase feed solid content (40-60%)
2. Optimize atomization pressure (100-200 bar)
3. Adjust inlet/outlet temperature ratio
4. Modify cyclone design for better separation
d₃₂ = (σ/ρ)^0.6 * (μ/σ)^0.1 * (Q/ΔP)^0.4 (for pressure nozzle)
Where d₃₂ = Sauter mean diameter
Optimization Parameters:
1. Increase feed solid content (40-60%)
2. Optimize atomization pressure (100-200 bar)
3. Adjust inlet/outlet temperature ratio
4. Modify cyclone design for better separation
25. Derive the equation for osmotic pump drug delivery system.
dm/dt = (A/h) * L_p * σ * Δπ * C_d
Where:
• dm/dt = drug release rate
• A = membrane area
• h = membrane thickness
• L_p = permeability
• σ = reflection coefficient
• Δπ = osmotic pressure difference
• C_d = drug concentration in core
Zero-order release achieved when Δπ is constant.
Where:
• dm/dt = drug release rate
• A = membrane area
• h = membrane thickness
• L_p = permeability
• σ = reflection coefficient
• Δπ = osmotic pressure difference
• C_d = drug concentration in core
26. Solve for filter area using Darcy's law: Q = (KAΔP)/(μL)
A = (QμL)/(KΔP)
Where:
Q = flow rate, μ = viscosity, L = cake thickness
K = permeability, ΔP = pressure drop
Critical for sizing filtration equipment in API manufacturing.
Where:
Q = flow rate, μ = viscosity, L = cake thickness
K = permeability, ΔP = pressure drop
Critical for sizing filtration equipment in API manufacturing.
27. Case Study: Roller compaction shows ribbon density variation. Provide mathematical analysis.
Johanson's Theory Analysis:
Pressure gradient: dP/dx = (μK P tanθ)/h
Where: μ = friction, K = material property, θ = nip angle, h = gap
Solutions:
1. Optimize feed screw speed/roll speed ratio
2. Control roll pressure profile
3. Adjust gap setting (1-3 mm typical)
4. Use force feeder for uniform feed
Pressure gradient: dP/dx = (μK P tanθ)/h
Where: μ = friction, K = material property, θ = nip angle, h = gap
Solutions:
1. Optimize feed screw speed/roll speed ratio
2. Control roll pressure profile
3. Adjust gap setting (1-3 mm typical)
4. Use force feeder for uniform feed
28. Calculate mixing index using Lacey's formula: M = (σ₀² - σ²)/(σ₀² - σ_r²)
Where:
σ₀² = variance of completely segregated mix
σ² = variance of actual mix
σ_r² = variance of random mix
M = 0 → completely segregated
M = 1 → perfectly random
σ₀² = variance of completely segregated mix
σ² = variance of actual mix
σ_r² = variance of random mix
M = 0 → completely segregated
M = 1 → perfectly random
29. Case Study: Fluid bed granulation shows overwetting. Provide droplet penetration time analysis.
Mathematical Analysis:
Droplet penetration time: t_p = (1.35V²/³)/(rε cosθ) * (μ/γ)
Where: V = droplet volume, r = pore radius, ε = porosity, θ = contact angle
Process Adjustments:
1. Reduce spray rate (10-50 mL/min per kg)
2. Increase inlet air temperature
3. Optimize nozzle position and atomization pressure
4. Adjust binder concentration (5-15%)
Droplet penetration time: t_p = (1.35V²/³)/(rε cosθ) * (μ/γ)
Where: V = droplet volume, r = pore radius, ε = porosity, θ = contact angle
Process Adjustments:
1. Reduce spray rate (10-50 mL/min per kg)
2. Increase inlet air temperature
3. Optimize nozzle position and atomization pressure
4. Adjust binder concentration (5-15%)
30. Derive the equation for heat transfer during drying: q = hA(T_air - T_surface)
Combined with mass transfer: m_dot = k_gA(p_s - p_air)
Where h = heat transfer coefficient, k_g = mass transfer coefficient
During constant rate period: m_dot = q/λ (λ = latent heat)
Where h = heat transfer coefficient, k_g = mass transfer coefficient
During constant rate period: m_dot = q/λ (λ = latent heat)
31. Solve for tablet ejection force: F_e = μ_s * F_c * exp(μ_k * θ)
Where:
μ_s = static friction coefficient
μ_k = kinetic friction coefficient
F_c = compression force
θ = angle of wrap
Lubricant reduces μ_s from ~0.5 to ~0.1
μ_s = static friction coefficient
μ_k = kinetic friction coefficient
F_c = compression force
θ = angle of wrap
Lubricant reduces μ_s from ~0.5 to ~0.1
32. Case Study: Wurster coating shows spray drying. Provide dimensionless number analysis.
Dimensionless Analysis:
Weber Number: We = (ρv²d)/σ (inertia vs surface tension)
Reynolds Number: Re = (ρvd)/μ (inertia vs viscosity)
Ohnesorge Number: Oh = √We/Re = μ/√(ρσd)
For proper coating: We > 10, Oh < 1
Adjustments: Increase atomization pressure, reduce viscosity, optimize droplet size.
Weber Number: We = (ρv²d)/σ (inertia vs surface tension)
Reynolds Number: Re = (ρvd)/μ (inertia vs viscosity)
Ohnesorge Number: Oh = √We/Re = μ/√(ρσd)
For proper coating: We > 10, Oh < 1
Adjustments: Increase atomization pressure, reduce viscosity, optimize droplet size.
33. Calculate the Z-score for blend uniformity: Z = (x̄ - μ)/(σ/√n)
For USP blend uniformity:
n = 10 samples, criteria: RSD ≤ 5%, all individual 85-115%
Z > 2.0 indicates significant deviation from target.
n = 10 samples, criteria: RSD ≤ 5%, all individual 85-115%
Z > 2.0 indicates significant deviation from target.
34. Case Study: Extrusion-spheronization yields dumbbell shapes. Provide rheological analysis.
Rheology Requirements:
Complex modulus: G* = √(G'² + G"²)
Loss tangent: tan δ = G"/G'
Optimal for extrusion: G' = 10³-10⁴ Pa, tan δ = 0.2-0.5
Formulation Adjustments:
1. Adjust water content (±5%)
2. Change MCC grade
3. Add binder (HPMC, PVP)
4. Optimize spheronization time (2-10 min)
Complex modulus: G* = √(G'² + G"²)
Loss tangent: tan δ = G"/G'
Optimal for extrusion: G' = 10³-10⁴ Pa, tan δ = 0.2-0.5
Formulation Adjustments:
1. Adjust water content (±5%)
2. Change MCC grade
3. Add binder (HPMC, PVP)
4. Optimize spheronization time (2-10 min)
35. Derive the Prandtl mixing length theory for turbulent flow in pipes.
τ_turb = ρ l² (du/dy)²
Where l = mixing length
u = velocity, y = distance from wall
Important for scale-up of mixing and transport processes.
Where l = mixing length
u = velocity, y = distance from wall
Important for scale-up of mixing and transport processes.
36. Solve for capsule fill weight variation: σ_total² = σ_powder² + σ_machine²
Where σ_powder depends on flow properties:
σ_powder ∝ 1/√(flowability)
To achieve RSD < 3%: Need Carr's Index < 15, Hausner Ratio < 1.25
σ_powder ∝ 1/√(flowability)
To achieve RSD < 3%: Need Carr's Index < 15, Hausner Ratio < 1.25
37. Case Study: Hot stage microscopy shows polymorph conversion during wet granulation.
Thermodynamic Analysis:
ΔG = ΔH - TΔS
Solvent-mediated conversion occurs when:
ΔG_solution(form II → form I) < 0
Prevention Strategies:
1. Use non-solvent granulating liquid
2. Control temperature during drying
3. Add polymer inhibitor
4. Select stable polymorph initially
ΔG = ΔH - TΔS
Solvent-mediated conversion occurs when:
ΔG_solution(form II → form I) < 0
Prevention Strategies:
1. Use non-solvent granulating liquid
2. Control temperature during drying
3. Add polymer inhibitor
4. Select stable polymorph initially
38. Calculate the compression work using Heckel analysis: ln[1/(1-D)] = KP + A
Where D = relative density, P = pressure, K = yield pressure
K = 1/3σ_y (σ_y = yield strength)
Typical values: MCC K = 100-200 MPa, lactose K = 200-400 MPa
K = 1/3σ_y (σ_y = yield strength)
Typical values: MCC K = 100-200 MPa, lactose K = 200-400 MPa
39. Case Study: Nanosuspension shows Ostwald ripening. Provide mathematical modeling.
Kelvin Equation Analysis:
S(r) = S(∞) * exp(2γV_m/rRT)
Where γ = interfacial tension, V_m = molar volume
Larger particles grow at expense of smaller ones.
Stabilization Strategies:
1. Add ripening inhibitor (same solubility)
2. Increase viscosity
3. Use polymeric stabilizers
4. Control storage temperature
S(r) = S(∞) * exp(2γV_m/rRT)
Where γ = interfacial tension, V_m = molar volume
Larger particles grow at expense of smaller ones.
Stabilization Strategies:
1. Add ripening inhibitor (same solubility)
2. Increase viscosity
3. Use polymeric stabilizers
4. Control storage temperature
40. Derive the equation for terminal velocity in cyclone separator.
v_t = (ρ_p - ρ_g)d²ω²r/(18μ)
Where ω = angular velocity, r = radius
Cut size: d_cut = √[9μW/(πN_e v_t (ρ_p - ρ_g))]
Where W = inlet width, N_e = effective turns
Where ω = angular velocity, r = radius
Cut size: d_cut = √[9μW/(πN_e v_t (ρ_p - ρ_g))]
Where W = inlet width, N_e = effective turns
41. Solve for disintegration time using Hixon-Crowell cube root law applied to disintegrants.
W₀¹/³ - W¹/³ = kt
Where W = tablet weight remaining
For superdisintegrants: k is 2-5 times higher than starch
Where W = tablet weight remaining
For superdisintegrants: k is 2-5 times higher than starch
42. Case Study: Bi-layer tablet shows layer separation. Provide stress analysis.
Stress Analysis:
Interfacial shear stress: τ = μσ_n
Where μ = interlayer friction, σ_n = normal stress
Solutions:
1. Optimize first layer roughness
2. Use adhesive interlayer
3. Adjust compression sequence
4. Control moisture gradient
Interfacial shear stress: τ = μσ_n
Where μ = interlayer friction, σ_n = normal stress
Solutions:
1. Optimize first layer roughness
2. Use adhesive interlayer
3. Adjust compression sequence
4. Control moisture gradient
43. Calculate the nucleation rate in crystallization: B⁰ = A exp(-ΔG*/kT)
Where ΔG* = 16πγ³v²/(3(kT lnS)²)
γ = interfacial energy, v = molecular volume, S = supersaturation
Control supersaturation to control crystal size distribution.
γ = interfacial energy, v = molecular volume, S = supersaturation
Control supersaturation to control crystal size distribution.
44. Derive the equation for diffusion through polymer membrane: J = (DKΔC)/h
Where D = diffusion coefficient, K = partition coefficient
ΔC = concentration difference, h = membrane thickness
Basis for controlled release formulations.
ΔC = concentration difference, h = membrane thickness
Basis for controlled release formulations.
45. Case Study: Coating solution shows orange peel effect. Provide dimensionless analysis.
Analysis:
Capillary Number: Ca = μv/γ (viscous vs surface forces)
For smooth film: Ca > 1, Oh < 0.1
Solutions:
1. Reduce viscosity (add solvent)
2. Increase spray rate
3. Optimize atomization
4. Adjust pan speed
Capillary Number: Ca = μv/γ (viscous vs surface forces)
For smooth film: Ca > 1, Oh < 0.1
Solutions:
1. Reduce viscosity (add solvent)
2. Increase spray rate
3. Optimize atomization
4. Adjust pan speed
46. Solve for gelation time using Winter-Chambon criterion: G' = G" at ω₀
Where ω₀ = crossover frequency
Gel point: tan δ = 1 at all frequencies
Critical for in-situ gelling formulations.
Gel point: tan δ = 1 at all frequencies
Critical for in-situ gelling formulations.
47. Calculate the Flory-Huggins interaction parameter for polymer-drug compatibility.
χ = (δ₁ - δ₂)²V₁/RT
Where δ = solubility parameter, V = molar volume
χ < 0.5 → miscible, χ > 0.5 → immiscible
Where δ = solubility parameter, V = molar volume
χ < 0.5 → miscible, χ > 0.5 → immiscible
48. Case Study: Rotary tablet press shows feeding issues. Provide force balance analysis.
Force Analysis:
Feeding force must overcome: F_friction + F_compaction
F_feed > μmg + F_compression
Solutions:
1. Optimize feed frame design
2. Improve powder flow (glidants)
3. Adjust paddle speed
4. Control fill depth
Feeding force must overcome: F_friction + F_compaction
F_feed > μmg + F_compression
Solutions:
1. Optimize feed frame design
2. Improve powder flow (glidants)
3. Adjust paddle speed
4. Control fill depth
49. Derive the equation for bubble point test in filter integrity testing.
P = 4γ cosθ/d
Where d = pore diameter, γ = surface tension
θ = contact angle (0° for hydrophilic filters)
For 0.22μm filter: P ≈ 50 psi
Where d = pore diameter, γ = surface tension
θ = contact angle (0° for hydrophilic filters)
For 0.22μm filter: P ≈ 50 psi
50. Solve for mixing efficiency using Danckwerts' intensity of segregation.
I = σ²/σ₀²
Where σ² = variance of sample, σ₀² = variance of segregated
I → 0 as mixing improves
Where σ² = variance of sample, σ₀² = variance of segregated
I → 0 as mixing improves
