Ideal Gas Law

Note: The relationship is a good approximation for gases at low density and at temperatures well above boiling point (i.e., far from phasing into a liquid).

$p V = N k_{B} T$

Where:
$p$ - absolute pressure
$V$ - volume that it fills
$N$ - number of molecules
$k_{B}$ - Boltzmann constant, $1.38 \times 10^{- 23} J / K$
$T$ - absolute temperature

Note: if the number of molecules does not change, then $p, V,$ and $T$ are only variables of concern.

Ideal Gas Law in Mole

Note: what you typically see in Chemistry

$p V = n R T$

$p$ - absolute pressure
$V$ - volume that it fills
$n$ - number of moles
$R$ - universal gas constant, $8.31 \frac{J}{m o l \cdot K}$
$T$ - absolute temperature

Van der Waals Equation of State

Note: modification for higher density gases
Note: $a$ and $b$ are different for each gas.
$[p + a (\frac{n}{V})^{2}] (V - n b) = n R T$

Internal Energy of Ideal Gas

Monotonic Gas

Note: Only energy for monotonic gas is translational energy.

$E_{i n t} = \frac{3}{2} N k_{B} T$

and mole equivalent

$E_{i n t} = \frac{3}{2} n R T$

Heat Capacity for Ideal Gases

Heat Capacity Based on Moles

Constant Volume

Note: for monatomic gases
$Q = n C_{v} Δ T$

$Q$ - heat transfer
$n$ - number of moles
$C_{v}$ - molar heat capacity at constant volume
$Δ T$ - change in temperature

Note: work cannot be done and the only change in internal energy is by heat transfer

$E_{i n t} = Q$

Recall $E_{i n t} = \frac{3}{2} n R T$ above:

$Δ E_{i n t} = \frac{3}{2} n R Δ T$

$Q = \frac{3}{2} n R Δ T$

Recall $Q = n C_{v} Δ T$ above:

$n C_{v} Δ T = \frac{3}{2} n R Δ T$ $

$C_{v} = \frac{3}{2} R$ ; independent of temperature