Ideal Gas Equation: MDCAT Chemistry notes
Ideal Gas Equation for MDCAT: PV = nRT from Boyle's, Charles's and Avogadro's laws, density and molar mass formulas, and worked problems.
Deriving the equation
The ideal (general) gas equation combines three gas laws:
- Boyle's law: $V\propto\frac{1}{P}$ (constant $T$, $n$)
- Charles's law: $V\propto T$ (constant $P$, $n$)
- Avogadro's law: $V\propto n$ (constant $P$, $T$)
Together $V\propto\frac{nT}{P}$, so
$$PV=nRT$$
where $R$ is the general gas constant and $T$ is the absolute temperature. The individual gas laws are all stated for a fixed amount of gas, so the number of moles is the variable they assume constant. For a fixed amount of gas under changing conditions:
$$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$
Key formulas
| Quantity | Formula | Obtained from |
|---|---|---|
| Moles | $n=\frac{PV}{RT}$ | Rearranging |
| Molar mass | $M=\frac{mRT}{PV}$ | Putting $n=m/M$ |
| Density | $d=\frac{PM}{RT}$ | Putting $d=m/V$ |
| Concentration (mol per volume) | $\frac{n}{V}=\frac{P}{RT}$ | Dividing by $V$ |
What density depends on
From $d=\frac{PM}{RT}$, the density of a gas depends on pressure, temperature and molar mass:
- Increasing pressure at constant temperature increases density.
- Increasing temperature at constant pressure decreases density.
- At the same $T$ and $P$, the gas with the lowest molar mass has the lowest density. Among Ne (20), $N_2$ (28), $O_2$ (32) and $F_2$ (38), neon is least dense.
Worked examples
Example 1. Density of $O_2$ at 1520 torr: 1520 torr = 2 atm, so $d=\frac{2\times32}{RT}=\frac{64}{RT}$.
Example 2. Both $T$ and $V$ doubled: $P=\frac{nRT}{V}$ is unchanged.
Example 3. 273 mL at STP (760 mm, 273 K) taken to 27 °C (300 K) and 600 mm: $V_2=273\times\frac{760}{600}\times\frac{300}{273}=380$ mL.
Example 4. To stop a gas expanding when more of it (more moles) is added, $V=\frac{nRT}{P}$ shows you must lower the temperature and increase the pressure.
Example 5. Moles of gas in 44.8 dm³ at 1 atm and 273 K: $n=\frac{1\times44.8}{0.0821\times273}=2.0$ mol.
Dalton's law of partial pressures
In a mixture of non-reacting gases, total pressure is the sum of partial pressures, and each partial pressure = mole fraction × total pressure. If a mixture at 50 atm contains 67.8% $H_2$ by moles, $p_{H_2}=\frac{67.8\times50}{100}$ atm.
Common MDCAT traps
- $T$ must be in kelvin; $PV=nRt$ with °C is wrong.
- Match units of $R$ to $P$ and $V$: use 0.0821 with atm and dm³, 8.314 with Pa and m³.
- $M=\frac{mRT}{PV}$, not $\frac{PV}{mRT}$.
- $PV=nRT$ is the general gas equation, not Dalton's or van der Waals' equation.
Quick revision
- $PV=nRT$ combines Boyle's, Charles's and Avogadro's laws.
- $d=PM/RT$.
- Higher pressure means higher gas density.
- $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$ for fixed $n$.
- Partial pressure = mole fraction × total pressure.