Ideal Gas Equation
- The equation of state for an ideal gas (or the ideal gas equation) can be expressed as:
$
\mathbf{p V}=\mathbf{n R T}
$
- The ideal gas equation can also be written in the form:
$
p \mathbf{V}=\mathbf{N k}
$
- An ideal gas is therefore defined as:
A gas which obeys the equation of state $p V=n R T$ at all pressures, volumes and temperatures
Worked example
A storage cylinder of an ideal gas has a volume of $8.3 \times 10^3 \mathrm{~cm}^3$. The gas is at a temperature of $15^{\circ} \mathrm{C}$ and a pressure of $4.5 \times 10^7 \mathrm{~Pa}$. Calculate the amount of gas in the cylinder, in moles.
Answer/Explanation
Step 1: Write down the ideal gas equation
Since the number of moles $(n)$ is required, use the equation:
$
\mathbf{p V}=\mathbf{n R} \mathbf{T}
$
Step 2:
Rearrange for the number of moles $\mathrm{n}$
$
\mathrm{n}=\frac{p V}{R T}
$
Step 3:
Substitute in values
$
\begin{gathered}
V=8.3 \times 10^3 \mathrm{~cm}^3=8.3 \times 10^3 \times 10^{-6}=8.3 \times 10^{-3} \mathrm{~m}^3 \\
T=15^{\circ} \mathrm{C}+273.15=288.15 \mathrm{~K} \\
\mathrm{n}=\frac{4.5 \times 10^7 \times 8.3 \times 10^{-3}}{8.31 \times 288.15}=155.98=160 \mathrm{~mol}(2 \mathrm{~s} . \mathrm{f} .)
\end{gathered}
$
Exam Tip
Don’t worry about remembering the values of R and k, they will both be given in the equation sheet in your exam.
The Boltzmann Constant
- The Boltzmann constant $k$ is used in the ideal gas equation and is defined by the equation:
$
\mathrm{k}=\frac{R}{N_A}
$
- Where:
$
\begin{aligned}
& \text { – } \mathrm{R}=\text { molar gas constant } \\
& \text { – } \mathrm{N}_{\mathrm{A}}=\text { Avogadro’s constant }
\end{aligned}
$
- Boltzmann’s constant therefore has a value of
$
k=\frac{8.31}{6.02 \times 10^{23}}=1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}
$
- The Boltzmann constant relates the properties of microscopic particles (e.g. kinetic energy of gas molecules) to their macroscopic properties (e.g. temperature)
- This is why the units are $\mathrm{J}^{-1}$
- Its value is very small because the increase in kinetic energy of a molecule is very small for every incremental increase in temperature