The Ideal Gas Equation
The grand synthesis of Boyle's, Charles's, and Avogadro's Laws.
Up to this point, we have studied the individual gas laws, which keep certain parameters constant to observe the relationship between the others. What happens if all variables ($P, V, T,$ and $n$) change simultaneously? We combine them into a single, elegant equation of state: The Ideal Gas Equation.
1. Derivation of the Equation
Let's combine the three fundamental volume relationships:
- Boyle's Law: $V \propto \frac{1}{P}$ (at constant $T, n$)
- Charles's Law: $V \propto T$ (at constant $P, n$)
- Avogadro's Law: $V \propto n$ (at constant $P, T$)
Combining these three proportionalities gives:
To remove the proportionality sign, we introduce a constant of proportionality, $R$ (the Universal Gas Constant):
This is the Equation of State for an Ideal Gas.
2. The Universal Gas Constant ($R$)
The value of $R$ depends entirely on the units of Pressure and Volume you plug into the equation. Using the wrong $R$ value is the #1 reason students lose marks in physical chemistry!
| Value of $R$ | Units of $R$ | When to use it? |
|---|---|---|
| 0.0821 | $\text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ | When Volume is in Liters and Pressure is in atm. |
| 8.314 | $\text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ | SI Unit: When Volume is in $m^3$ and Pressure is in $Pa$ (or energy is in Joules). |
| 0.08314 | $\text{L} \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ | When Volume is in Liters and Pressure is in bar. |
| 1.987 $\approx$ 2 | $\text{cal} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ | When dealing with thermodynamics problems requiring Calories. |
3. Density and Molar Mass Relationship
We can modify the Ideal Gas Equation to find the density ($d$) or the Molar Mass ($M$) of an unknown gas.
Substitute moles ($n$) with $\frac{\text{mass }(m)}{\text{Molar Mass }(M)}$:
Rearrange to group mass and volume ($d = \frac{m}{V}$):
The "PM = dRT" Equation:
$$PM = dRT$$
4. The Combined Gas Law
If the amount of gas (moles, $n$) is kept constant, but $P, V,$ and $T$ all change, we can rearrange $PV = nRT$ to put all constants on one side:
This gives us the Combined Gas Law, used to find a new state when an initial state changes completely:
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