Molar Volume \(V_m\) and Standard Conditions
For an ideal gas, the molar volume (volume per mole) at temperature \(T\) and pressure \(P\) is
\( V_m \;=\; \dfrac{RT}{P} \)
From the ideal–gas law
\[
PV = nRT \quad\Longrightarrow\quad
\frac{V}{n} = \frac{RT}{P} \equiv V_m
\]
Here \(R=8.314\,462\,618\ \mathrm{J\,mol^{-1}\,K^{-1}} = 8.314\,462\,618\ \mathrm{Pa\,m^{3}\,mol^{-1}\,K^{-1}}\).
Using SI gives \(V_m\) in \(\mathrm{m^{3}\,mol^{-1}}\) (convert to L·mol⁻¹ by multiplying by \(10^{3}\)).
Common reference states
- STP (IUPAC): \(T=273.15\ \mathrm{K}\), \(P=1\ \mathrm{bar}\) → \(V_m \approx 22.711\ \mathrm{L\,mol^{-1}}\).
- 0 °C & 1 atm: \(T=273.15\ \mathrm{K}\), \(P=1\ \mathrm{atm}\) → \(V_m \approx 22.414\ \mathrm{L\,mol^{-1}}\).
- SATP: \(T=298.15\ \mathrm{K}\), \(P=1\ \mathrm{bar}\) → \(V_m \approx 24.789\ \mathrm{L\,mol^{-1}}\).
- RTP: \(T=298.15\ \mathrm{K}\), \(P=1\ \mathrm{atm}\) → \(V_m \approx 24.465\ \mathrm{L\,mol^{-1}}\).
Values above follow \(V_m=\dfrac{RT}{P}\) with \(R\) in SI; rounding shown to 3–4 s.f.
Using \(V_m\) to convert between \(V\) and \(n\)
\[
V = n\,V_m
\qquad\qquad
n = \frac{V}{V_m}
\]
- Always convert to \(T\) in kelvin and \(P\) in pascals before computing \(V_m\).
- Work in SI (m³, mol), then convert results (e.g., to L or mmol) as desired.
Example 1 — volume at STP
Given: \(n=1.00\ \mathrm{mol}\) at STP \((273.15\ \mathrm{K},\ 1\ \mathrm{bar})\).
\[
V_m=\frac{RT}{P}
= \frac{8.314462618\times 273.15}{1.0000\times 10^{5}}
= 2.2711\times 10^{-2}\ \mathrm{m^{3}}
= 22.711\ \mathrm{L}
\]
\[
V = n\,V_m = 1.00\times 22.711 = 22.711\ \mathrm{L}
\]
Example 2 — moles at RTP
Given: \(V=12.0\ \mathrm{L}\) at RTP \((298.15\ \mathrm{K},\ 1\ \mathrm{atm})\).
\[
V_m=\frac{8.314462618\times 298.15}{101325}
= 2.4465\times 10^{-2}\ \mathrm{m^{3}}
= 24.465\ \mathrm{L\,mol^{-1}}
\qquad\Rightarrow\qquad
n=\frac{12.0\ \mathrm{L}}{24.465\ \mathrm{L\,mol^{-1}}}
\approx 0.491\ \mathrm{mol}
\]
Notes & pitfalls
- Ideal behavior. At high pressures or very low temperatures real gases deviate; \(V_m\) then differs from \(RT/P\).
- Units. Mixing atm/bar/kPa without converting to Pa leads to wrong \(V_m\). Keep \(T\) in K and \(P\) in Pa for the formula.
- Graph. At fixed \(T,P\), a plot of \(V\) versus \(n\) is a straight line through the origin with slope \(V_m\).
