Science & Education

pH directly from concentration: strong solutes fully dissociate, weak solutes solve the exact quadratic with α and the 5% rule.

Health & Psychology

Score PHQ-9, GAD-7 and PCL-5 self-report screens with official bands, the PHQ-9 item-9 suicide-risk flag and depression algorithm, and the PCL-5 DSM-5 cluster rule.

Science & Education

B_λ(T), Wien peak λ_max = b/T, M_λ = πB_λ at the peak and at any wavelength, and the Stefan-Boltzmann total σT⁴.

Science & Education

I = I₀·cos²θ: transmitted intensity and percentage for any polarizer angle, with the 0°/45°/90° landmarks.

Math & Numbers

σ(n) = 2n ⟺ perfect. Factorization-based σ(n), proper-divisor sum, deficient/perfect/abundant classification, abundancy index, Euclid–Euler form. n ≤ 10¹².

Science & Education

E_k(max) = hν − φ with preset metal work functions or custom φ: threshold frequency, cutoff wavelength, emission verdict, stopping voltage, and photoelectron speed.

Math & Numbers

Three-point PERT estimates: te and σ² per activity, critical path, expected project duration and variance, and normal completion probability for a target time.

Math & Numbers

Convert pH ↔ [H⁺] in mol/L, mmol/L, µmol/L or nmol/L with the full unit table — pH 7.40 ↔ 39.8 nmol/L.

Health

Compute phage burst size = (PFU after lysis − PFU input) / infected cells from a one-step growth experiment, with Poisson MOI guidance and typical-range interpretation.

Health

Compute PFU/mL titer from plaque counts, dilution factor, and plated volume, with multi-plate statistics and 30–300 counting-range warnings.

Media

Build every podcast chapter format from one timecoded list: Podcasting 2.0 JSON + RSS tag, ID3v2.4 CHAP+CTOC burned into an MP3, Vorbis comments, mp4chaps, YouTube timestamps and SRT, with a per-player support matrix.

Math & Numbers

Convert magnetic permeability (μ) between absolute permeability in henry per metre (H/m, SI unit; equivalently N/A² or T·m/A) and relative permeability μ_r (dimensionless). Vacuum permeability μ₀ = 4π×10⁻⁷ H/m ≈ 1.25663706×10⁻⁶ H/m (defined pre-2019; since the 2019 SI redefinition it is a measured quantity with relative uncertainty 1.6×10⁻¹⁰, so the 4π×10⁻⁷ form is still exact for all engineering purposes). Relations used: μ = μ_r × μ₀ and μ_r = μ / μ₀ (magnetic susceptibility χ_m = μ_r − 1). Reference: vacuum μ_r = 1, copper 0.999994 (diamagnetic), aluminum 1.000022 (paramagnetic), iron ≈ 5000 (ferromagnetic).