Course-Wide Conventions & Notation

This page is a one-page reference for the conventions used throughout CHEM 4020/5020.

Sign conventions (heat $q$ and work $w$)¶

We will use the chemistry sign convention throughout:

• Heat $q$ (or $\delta q$):

  • $q>0$ when heat is absorbed by the system.

  • $q<0$ when heat is released by the system.

• Work $w$ (or $\delta w$):

  • $w>0$ when work is done on the system.

  • $w<0$ when work is done by the system.

With this convention, the First Law reads

$P\,dV$ work (most common case)¶

For pressure–volume work against an external pressure $P_{\mathrm{ext}}$,

Therefore:

• Expansion ($dV>0$) gives $\delta w<0$ (system does work on surroundings).

• Compression ($dV<0$) gives $\delta w>0$ (surroundings do work on system).

For a reversible $PV$ process, $P_{\mathrm{ext}} = P$, so we often write $\delta w_{\mathrm{rev}} = -P\,dV$.

Standard state and the “∘” symbol¶

A superscript ∘ denotes a standard-state quantity (e.g., $\mu^\circ$, $\Delta G^\circ$, $\Delta H^\circ$, $S^\circ$, $K^\circ$).

In this course, for gases, we take the standard-state pressure to be

Notes:

• Standard-state quantities can still depend on temperature (e.g., $\Delta G^\circ(T)$).

• When we use activities, the standard state corresponds to unit activity ($a=1$).

Per-molecule vs. per-mole quantities ($k_{\mathrm{B}}$ vs. $R$)¶

We will be explicit about how we count:

• $N$ = number of molecules/particles

• $n$ = number of moles

The ideal-gas equation is written in either form:

The constants are related by

Rule of thumb:

• Use $k_{\mathrm{B}}$ for per-particle (stat mech / microscopic) quantities.

• Use $R$ for per-mole (thermo / macroscopic) quantities.

Reserved symbols and notation choices¶

Some symbols vary across textbooks. To minimize collisions, we adopt the following defaults.

(A) $Z$ is reserved for compressibility¶

• Compressibility factor:

  $$Z \equiv \frac{PV}{nRT}.$$

  (6)

  • $Z=1$ for an ideal gas.

  • $Z\neq 1$ indicates non-ideal behavior.

We do not use $Z$ to denote a partition function.

(B) Partition functions use $q$ and $Q$¶

• $q$ = one-particle / one-molecule partition function

• $Q$ = many-particle (canonical) partition function

(Many texts use $Z$ for a partition function; we will not.)

(C) Chemical equilibrium uses $Q$ and $K$¶

• $Q$ (often written $Q_p$, $Q_c$, etc.) = reaction quotient

• $K$ = equilibrium constant

When the same letter appears in different contexts (e.g., $Q$ as a partition function vs. $Q$ as a reaction quotient), we will rely on context and/or subscripts ($Q(T,V,N)$ vs. $Q_p$, $Q_c$) to keep the meaning clear.

Quick lookup table¶