Definite Integrals in Chemistry

A definite integral is just a number — but in chemistry that number is always something physical: an amount of work done, a probability, or an average. The Fundamental Theorem tells us to evaluate it by finding the antiderivative $F$ and subtracting: ${\int }_a^{b}f(x)dx=F(b)-F(a)$.

In thermodynamics, $W={\int }_{V_1}^{V_2}pdV$ is the work of expansion — the area under the pressure–volume curve. For an ideal gas with $p=nRT/V$, integrating $1/V$ produces a logarithm and gives $W=nRT\ln (V_2/V_1)$.

In quantum mechanics, ${\int }_0^L|{\psi }_n{|}^{2}dx=1$ is the normalisation condition — the total probability of finding the particle somewhere must be exactly one. And $⟨x⟩={\int }_0^L{\psi }^{\ast }x\psi dx$ is the expectation value of position — the quantum-mechanical average that experiments ultimately measure.

Entropy changes also live here: $\Delta S={\int }_{T_1}^{T_2}\frac{C_p}{T}dT$ integrates another $1/T$ (yielding a logarithm again) to find how entropy changes with temperature.

The average value of any function over an interval $[a,b]$ is the definite integral divided by the interval length: $\bar{f}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$. In spectroscopy this is how the time-averaged dipole moment is computed; in thermodynamics it is how average properties over a temperature range are tabulated. The expectation-value formula $⟨O⟩=\int {\psi }^{\ast }\hat{O}\psi dx$ is a weighted version of exactly this average — the weighting function is the probability density $|\psi {|}^2$.

Work integrals highlight an important sign convention: chemists define $w$ as work done on the system, so the reversible expansion formula carries a negative sign, $w=-nRT\ln (V_2/V_1)$. When using the physics convention (work done by the system), the sign is positive. Always state which convention you are using.