# What are all the relevant extensive and intensive thermodynamic quantities?

## I am listing these for public reference purposes! :)

Nov 19, 2015

The most commonly taught Thermodynamic quantities are:

INTENSIVE QUANTITIES
Intensive quantities/properties do NOT depend on the amount of substance there is.

Chemical Potential

${\mu}_{j} = {\mu}_{j}^{\text{*}} + R T \ln {\chi}_{j}$
where ${\chi}_{j} = \frac{{n}_{j}}{{n}_{\text{tot}}}$, ${n}_{j} = \text{moles of substance j}$, and $\text{*}$ means "without solute".

Density

$\rho = \frac{m}{V}$

Pressure $P$
Temperature $T$

Constant-Pressure Specific Heat Capacity

${C}_{P} = {\left(\frac{\partial H}{\partial T}\right)}_{P}$
(the first derivative of the enthalpy with respect to temperature at a constant pressure)

Constant-Volume Specific Heat Capacity

${C}_{V} = {\left(\frac{\partial U}{\partial T}\right)}_{V}$
(the first derivative of the internal energy with respect to temperature at a constant volume)

You have used ${C}_{P}$ before in General Chemistry within the equation:

${q}_{P} = m {C}_{P} \Delta T$

EXTENSIVE QUANTITIES
Extensive quantities/properties do depend on the amount of substance there is.

Mass $m$
Volume $V$

Path Functions (dependent on path taken in order to get from initial state to final state):

Heat flow

$q \ge T \mathrm{dS}$
${q}_{\text{rev}} = T \mathrm{dS}$

Pressure-volume work

$w = - \int P \mathrm{dV}$

State Functions (dependent only on initial and final states)

Internal Energy

$\textcolor{b l u e}{\mathrm{dU} = T \mathrm{dS} - P \mathrm{dV}}$

$\textcolor{b l u e}{\Delta U = q + w}$

Enthalpy
Enthalpy is the amount of heat transferred due to heat flow $\left(q\right)$ and PV work that involves a change in the pressure.

$\textcolor{b l u e}{\mathrm{dH} = T \mathrm{dS} + V \mathrm{dP}}$

$\textcolor{b l u e}{\Delta H = \Delta U + \Delta \left(P V\right)}$
$= q + w + P \Delta V + V \Delta P + \Delta P \Delta V$
$= T \Delta S - P \Delta V + P \Delta V + V \Delta P + \Delta P \Delta V$
$= \textcolor{b l u e}{T \Delta S + V \Delta P + \Delta P \Delta V}$

You often work at a constant pressure in General Chemistry where $\Delta H = {q}_{P}$.

Entropy
Entropy is the amount of disorder in the system, and is equal to the amount of reversible heat flow possible per unit of temperature. In other words, it is the capacity to experience an increase in motion due to a change in temperature.

$\textcolor{b l u e}{\Delta S \ge 0}$
$\textcolor{b l u e}{\ge} \left[\int \frac{\partial q}{T} = \textcolor{b l u e}{\frac{q}{T}}\right]$
$= \int \left(\partial {q}_{\text{rev"))/T color(blue)(= (q_"rev}} / T\right)$

Gibbs' free energy
The Gibbs' free energy is commonly known as the maximum amount of process-initiating work that can be obtained from a closed system. This is commonly associated with constant pressure.

$\textcolor{b l u e}{\mathrm{dG} = - S \mathrm{dT} + V \mathrm{dP}}$

$\textcolor{b l u e}{\Delta G = \Delta H - T \Delta S}$ at constant temperature
$\text{ } = \textcolor{b l u e}{\Delta {G}^{\circ} + R T \ln Q}$ at constant temperature
$\text{ } = \textcolor{b l u e}{\Delta A + \Delta \left(P V\right)}$

Helmholtz free energy
The Helmholtz free energy is commonly known as the amount of process-initiating energy that can be obtained from a closed system. This is commonly associated with constant volume.

$\textcolor{b l u e}{\mathrm{dA} = - S \mathrm{dT} - P \mathrm{dV}}$

$\textcolor{b l u e}{\Delta A = \Delta U - T \Delta S}$ at constant temperature
$\text{ } = \textcolor{b l u e}{\Delta {A}^{\circ} + R T \ln Q}$ at constant temperature
$\text{ } = \textcolor{b l u e}{\Delta G - \Delta \left(P V\right)}$