My B.Sc. Study Space

🧪 Chemistry — Major

Unit VIII — Mathematical Concepts for Chemistry · Calculus & Probability for Chemists

Differentiation, Integration & Logarithms

CompleteFundamentals of Chemistry

The calculus toolkit chemists use for kinetics, thermodynamics and quantum chemistry.

Full notes

Definition: Differentiation finds the instantaneous rate of change (slope) of a function — how a property changes when another varies infinitesimally. Integration is the reverse: it sums infinitesimal contributions to give a total change.

1. Standard derivatives used in chemistry

  • d/dx (Kx) = K
  • d/dx (xⁿ) = nxⁿ⁻¹
  • d/dx (eˣ) = eˣ
  • d/dx (eᵏˣ) = keᵏˣ
  • d/dx (sin x) = cos x
  • d/dx (ln x) = 1/x

2. Maxima and minima

Set the first derivative to zero to find turning points (that's where reaction rates peak, potential wells bottom out, etc.). The second-derivative sign decides max (negative) or min (positive).

3. Integration of useful functions

  • ∫ dx = x + C
  • ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
  • ∫ 1/x dx = ln|x| + C
  • ∫ eᵏˣ dx = eᵏˣ/k + C
  • Definite integrals give areas — e.g., area under a concentration-time curve.

4. Logarithmic relations

ln(xy) = ln x + ln y; ln(x/y) = ln x − ln y; ln(xⁿ) = n ln x. These linearize exponential laws: ln[A] = ln[A]₀ − kt (first-order kinetics) plots as a straight line whose slope gives −k.

5. Slope of linear graphs

For y = mx + c, the slope m = Δy/Δx. In chemistry, plotting the right variable gives a line whose slope encodes a constant: Arrhenius plot ln k vs 1/T → slope = −Eₐ/R.

Note: Exam point: first-order rate law, Arrhenius equation and Gibbs energy all communicate through straight-line/log plots — master logarithms.

Exam notes

d/dx(xⁿ)=nxⁿ⁻¹; d/dx(eˣ)=eˣ; ∫1/x = ln; slope m=Δy/Δx; maxima at d/dx=0. ln-linearization gives slopes k and Eₐ/R from data.

Important points

  • This unit directly feeds Unit VIII of Sem 3 (kinetics) and Sem 4 (quantum mechanics).

Common mistakes

  • Forgetting the constant of integration in indefinite integrals.
PreviousHardware, Software, Number Systems & OSNext Permutations, Combinations & Probability