Pharma and Physics: Modeling Drug Diffusion and Jet Fuel Combustion with Conservation Laws

Hook: Real-world stories, real physics problems

Struggling to turn abstract physics into exam-ready calculations? You’re not alone. In 2026, headlines linking pharma developments and jet fuel worries (from faster drug-review programs to fuel performance questions) offer perfect, exam-style contexts for mastering diffusion, Fick’s laws, combustion, and the conservation laws of mass and energy. This article converts those stories into step-by-step, AP/college-level worksheets and worked examples you can solve, teach, or reuse in class.

What you’ll learn (inverted pyramid)

• How to set up mass and energy balances for drug diffusion and jet-fuel combustion.
• How to apply Fick’s first and second laws and estimate timescales with characteristic diffusion times.
• How to compute stoichiometry, air–fuel ratios, and release energy for hydrocarbon combustion.
• How to convert news-driven scenarios into exam-ready problems and quick checks.

2026 context and trends you should know

Late 2025 and early 2026 saw two related trends that matter in classroom modeling: (1) pharma companies increasingly pair laboratory experiments with high-fidelity numerical models (including physics-informed neural networks) to predict controlled-release behavior under regulatory pressure; (2) aviation research pushes improved combustion models for both conventional jet fuels and sustainable aviation fuels (SAFs), delivered by faster cloud-based CFD tools. These shifts mean physics students should not only master textbook equations but also learn how to select assumptions, estimate orders of magnitude, and validate models against real data.

Core concepts (brief refresher)

Conservation laws

Mass conservation: For any control volume, accumulation = inflow − outflow + generation (if chemical reaction creates mass) − consumption. In non-relativistic chemistry, mass is conserved; species may transform but total mass stays constant.

Energy conservation

First law of thermodynamics: ΔE = Q − W + energy carried in − energy carried out. For adiabatic closed combustion, chemical energy converts to sensible/internal energy of products (and radiation/losses if included).

Fick’s laws

• Fick’s first law (steady state): J = −D (dC/dx) — flux J (mol·m⁻²·s⁻¹) proportional to concentration gradient.
• Fick’s second law (transient): ∂C/∂t = D ∂²C/∂x² — describes how concentration evolves over time.

Useful timescale

Characteristic diffusion time: t_char ≈ L²/D. If t_char is much smaller than the experiment time, diffusion reaches near steady state. If larger, transient effects dominate.

Worked Example Set 1 — Pharma: Controlled-release patch (steady and transient)

Scenario (inspired by 2026 pharma modeling interest): A transdermal patch contains a drug reservoir kept at constant concentration C_s against a polymer membrane of thickness L. The skin side acts as a perfect sink (approx. zero concentration). Calculate the steady diffusive flux, estimate the time to reach steady state, and set up a lumped-compartment model for plasma concentration.

Given

• Membrane thickness L = 0.50 mm = 5.0×10⁻⁴ m
• Diffusion coefficient in membrane D = 1.0×10⁻¹⁰ m²·s⁻¹
• Reservoir concentration C_s = 100 mol·m⁻³
• Patch contact area A = 10 cm² = 1.0×10⁻³ m²
• Blood/plasma volume V = 5.0 L = 5.0×10⁻³ m³
• Elimination half-life t_1/2 = 4.0 h → k_el = ln2/4 h = 0.1733 h⁻¹

Problem 1a — Steady-state flux and mass delivery rate

Use Fick’s first law with constant gradient approximation (C changes linearly from C_s to 0 across L):

J = −D (ΔC/Δx) = D (C_s − 0)/L

Compute J:

J = (1.0×10⁻¹⁰ m²·s⁻¹)(100 mol·m⁻³)/(5.0×10⁻⁴ m) = 2.0×10⁻⁵ mol·m⁻²·s⁻¹

Mass delivery rate (moles per second) = J × A = 2.0×10⁻⁵ × 1.0×10⁻³ = 2.0×10⁻⁸ mol·s⁻¹

Convert to mol·hr⁻¹: multiply by 3600 → 7.2×10⁻⁵ mol·hr⁻¹.

Problem 1b — Characteristic diffusion time

t_char ≈ L²/D = (5.0×10⁻⁴ m)² / (1.0×10⁻¹⁰ m²·s⁻¹) = 2.5×10³ s ≈ 0.694 h ≈ 42 min.

Interpretation: The membrane diffusion timescale is under an hour, so steady-state flux is a reasonable approximation after ~a few t_char (~2–4× t_char).

Problem 1c — Lumped compartment model for plasma concentration

Set up mass balance for drug in blood (mol): V dC_b/dt = J A − k_el V C_b, where C_b(t) is blood concentration (mol·m⁻³), J A is mol·s⁻¹ input, and k_el is elimination constant in s⁻¹.

Convert k_el to s⁻¹: k_el = 0.1733 h⁻¹ = 4.81×10⁻⁵ s⁻¹. Input mol·s⁻¹ is 2.0×10⁻⁸ mol·s⁻¹.

Solve steady-state (dC_b/dt = 0): C_b,ss = (J A)/(k_el V)

C_b,ss = (2.0×10⁻⁸ mol·s⁻¹) / (4.81×10⁻⁵ s⁻¹ × 5.0×10⁻³ m³) = (2.0×10⁻⁸) / (2.405×10⁻⁷) ≈ 0.0832 mol·m⁻³.

Convert to mg·L⁻¹ if molar mass M_drug = 200 g·mol⁻¹: 0.0832 mol·m⁻³ × 200 g·mol⁻¹ = 16.64 g·m⁻³ = 16.64 mg·L⁻¹.

Takeaways for students

• Always check units: converting between s vs h and m³ vs L is a common exam trap.
• Use t_char = L²/D to decide if the steady-state approximation is valid.
• Conservation of mass drives the compartment ODE; be clear what your control volume contains.

Worked Example Set 2 — Jet fuel combustion: stoichiometry, mass balance, and energy

News-driven context: debates about jet-fuel safety and emissions in 2025–2026 push engineers to calculate stoichiometric air requirements and energy release for jet fuel and SAF blends. We'll use a common surrogate hydrocarbon, dodecane (C12H26), as a simple kerosene-like fuel.

Given

• Fuel surrogate: C12H26 (dodecane)
• Molar mass M_f = 12×12.011 + 26×1.008 ≈ 170.34 g·mol⁻¹ = 0.17034 kg·mol⁻¹
• Lower heating value (approx) LHV ≈ 7.51×10³ kJ·mol⁻¹ ≈ 44.1 MJ·kg⁻¹ (typical jet fuel ~43 MJ·kg⁻¹)
• Air composition ≈ 21% O₂ by mole; mean molar mass of air ≈ 28.97 g·mol⁻¹

Problem 2a — Stoichiometric O₂ and air per mole and per kg of fuel

Combustion stoichiometry for CnHm: CnHm + (n + m/4) O₂ → n CO₂ + (m/2) H₂O. For C12H26: O₂ required = 12 + 26/4 = 18.5 mol O₂ per mol fuel.

Moles of air required (stoichiometric) = 18.5 / 0.21 ≈ 88.1 mol air per mol fuel.

Mass of air per mole fuel = 88.1 mol × 28.97 g·mol⁻¹ = 2553 g = 2.553 kg air per 0.17034 kg fuel.

Air-to-fuel mass ratio ≈ 2.553 / 0.17034 ≈ 15.0 (≈ 15:1), consistent with typical kerosene stoichiometry.

Problem 2b — Energy release per kg fuel and approximate adiabatic flame temperature

Heat released Q_per_kg ≈ LHV ≈ 44.1 MJ·kg⁻¹.

For 1 kg fuel, total product mass ≈ fuel + air ≈ 1 + 15 = 16 kg (simplified, neglecting slight mass change from O/H atoms combining — mass is conserved, this is fuel+air mass).

Assuming the combustion is adiabatic and the product gases have average specific heat cp ≈ 1.15 kJ·kg⁻¹·K⁻¹ at high temperatures (simplified constant cp), adiabatic temperature rise ΔT ≈ Q/(m_products cp).

ΔT ≈ 44,100 kJ / (16 kg × 1.15 kJ·kg⁻¹·K⁻¹) ≈ 44,100 / 18.4 ≈ 2,397 K.

If ambient was 300 K, adiabatic flame temperature ≈ 2,700 K. Note: This is an estimate. Real combustors see lower temperatures due to dissociation, heat losses, and non-ideal mixing; modern CFD with chemical kinetics gives more accurate values.

Problem 2c — Conservation check: CO₂ produced per kg fuel

Moles CO₂ per mole fuel = n = 12 mol CO₂ per mol fuel. For 1 mol fuel (0.17034 kg), CO₂ mass = 12 × 44.01 g = 528.12 g ≈ 0.528 kg CO₂ per 0.17034 kg fuel → per kg fuel: 0.528 / 0.17034 ≈ 3.10 kg CO₂ per kg fuel.

Therefore, 1 kg jet fuel gives ≈ 3.1 kg CO₂ at complete combustion — a useful check when comparing emissions across fuels and SAF blends.

Takeaways for students

• Work systematically: write the balanced chemical equation first, then compute moles and masses.
• Use molar masses to convert between mol and kg.
• Energy per unit mass (MJ/kg) is often the most useful engineering metric.

Challenge problems (worksheet-ready)

1. Pharma diffusion — semi-infinite medium: A drug is released from a disk into tissue approximated as a semi-infinite medium. Given surface concentration held at C0 = 0.5 mol·m⁻³ and D = 8×10⁻¹¹ m²·s⁻¹, find the mass per unit area absorbed after 24 h. Hint: use M(t)/A = 2 C0 √(D t/π) for a constant surface concentration into a semi-infinite medium.
2. Jet fuel transient energy: A combustor injects 0.5 kg·s⁻¹ of kerosene (assume LHV = 43 MJ·kg⁻¹). If the combustor exhausts products at 900 K and the inlet air is 300 K, estimate the useful thermal power available assuming perfect heat capture (ignore pressure work). Use cp = 1.15 kJ·kg⁻¹·K⁻¹ and stoichiometric air-to-fuel of 15:1.
3. Combined diffusion–reaction: A drug diffuses across a membrane and is metabolized in the receiving compartment with first-order rate constant k = 0.2 h⁻¹. Set up the PDE/ODE coupling and derive the steady-state plasma concentration in terms of J, A, V, and k.

Advanced strategies and 2026 modeling tips

As modeling tools matured in 2025–26, three practices improved exam-style modeling and lab projects:

• Dimensional analysis first — nondimensional groups (e.g., Fourier number Fo = Dt/L²) tell you which terms dominate before you solve.
• Simplify then refine — start with steady approximations or lumped models; if predictions disagree with data by orders of magnitude, add detail (transient terms, kinetics, heat losses).
• Validate with quick experiments or literature values — many students benefit from checking diffusion coefficients and cp values against handbooks or open-source repositories and databases. In 2026, open-source repositories and community-shared CFD cases make validation easier.

Common exam pitfalls and how to avoid them

• Unit mismatch: Always put an explicit unit conversion step in your solution.
• Forgetting to state assumptions: e.g., adiabatic, perfect sink, steady vs transient — these determine correct equations.
• Not checking limits: Evaluate your solution for extreme cases (D→0, L→0) to see if the behavior makes sense.

Pro tip: On exams, write the balance equation in words first (inflow minus outflow plus generation) — markers reward clear reasoning as much as algebra.

Teacher notes and classroom uses

• Split the worked examples into 20–30 minute problem-solving stations: Station 1 (Fick’s law steady), Station 2 (transient estimate), Station 3 (stoichiometry), Station 4 (energy balance).
• Assign the challenge problems as a homework set, and ask students to submit both numeric answers and a short paragraph listing assumptions and possible sources of error.
• For advanced courses, pair these worksheet problems with a short Jupyter notebook that numerically solves the diffusion PDE (using finite differences) or a simple zero-dimensional reactor for combustion energetics.

Final checklist for solving similar problems

1. Define your control volume and list unknowns.
2. Write conservation equations (mass/species, energy) in symbolic form.
3. Choose steady or transient models and justify with timescales (t_char or Fo).
4. Plug numbers, carry units, and check limiting behavior.

Closing thoughts — why these problems matter in 2026

Physics problems grounded in pharma and jet-fuel stories teach more than equations: they train students to synthesize data, reason with assumptions, and communicate limits. In 2026, with pharma regulators and aviation engineers demanding faster and more accurate modeling, these skills are career-relevant. Whether you’re studying for AP, prepping for campus exams, or designing a class lab, the conservation laws and diffusion/combustion examples here connect textbook fundamentals with pressing real-world problems.

Actionable next steps

• Use the worked examples above to create a 50–60 minute classroom lab. Time breakdown: 15 min setup, 30 min calculations, 15 min discussion.
• Download a starter Jupyter notebook (search our 2026 resource page) that numerically solves the diffusion PDE for the patch case — run it on cloud VM if your laptop is slow.
• Practice one challenge problem per week and submit to a teacher or peer for feedback; focus on assumptions and unit-checking.

Call to action

If you found these worked problems useful, sign up for our free worksheet pack and Jupyter starter notebooks tailored to AP and college physics. Bring one real news story (pharma or aviation) to your next study session and turn it into a conservation-law problem — then tag us and we’ll share the best student solutions.

Ready to level up? Download the printable worksheet, run the notebook, and try the challenge problems today.

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