DAT General Chemistry: The Combined Gas Law

DAT High-Yield Concept

The Core Equation: The Combined Gas Law combines pressure, volume, and absolute temperature for a fixed amount of gas: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$ .
Absolute Temperature Rule: Temperature must always be converted to Kelvin ( $\pu{K} = \pu{degC} + 273.15$ ) prior to calculations.
Matching Units: Pressure and volume units must be identical on both sides, though any standard unit (e.g., $\pu{atm}$ , $\pu{mmHg}$ , $\pu{L}$ , $\pu{mL}$ ) is acceptable.
Special Cases: Holding any single state variable constant isolates Boyle's, Charles's, or Gay-Lussac's law.

The Combined Gas Law Equation

Combined Gas Law: For a closed system containing a fixed amount of gas (

n = \text{constant}

), the product of pressure and volume divided by absolute temperature remains constant:

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

When solving two-state gas problems where pressure, volume, and temperature simultaneously change, individual gas laws are insufficient. The combined gas law integrates Boyle's Law ( $P \propto 1/V$ ), Charles's Law ( $V \propto T$ ), and Gay-Lussac's Law ( $P \propto T$ ) into a single relationship derived from the Ideal Gas Law when molar quantity $n$ is held constant.

Two-State Problems with Three Changing Variables

DAT questions frequently present two-state scenarios where initial conditions (State 1: $P_1, V_1, T_1$ ) change to new conditions (State 2: $P_2, V_2, T_2$ ). To solve these problems efficiently under time constraints:

Identify the knowns and the single unknown variable.
Convert all temperatures to Kelvin: $T(\pu{K}) = T(\pu{degC}) + 273.15$ .
Ensure initial and final units for pressure and volume match.
Rearrange the combined gas law algebraically to isolate the unknown variable before plugging in numerical values.

Worked Example

Problem: A sealed flexible cylinder contains a sample of $\ce{N2}$ gas with an initial volume of $\pu{4.00 L}$ at $\pu{1.20 atm}$ and $\pu{27.0 degC}$ . The gas is compressed to $\pu{2.00 L}$ while being heated to $\pu{127.0 degC}$ . Calculate the final pressure of the gas in atmospheres.

Solution:

1. State 1 variables: $P_1 = \pu{1.20 atm}$ , $V_1 = \pu{4.00 L}$ , $T_1 = \pu{27.0 degC} + 273.15 = \pu{300.15 K}$ .

2. State 2 variables: $P_2 = ?$ , $V_2 = \pu{2.00 L}$ , $T_2 = \pu{127.0 degC} + 273.15 = \pu{400.15 K}$ .

3. Rearrange for $P_2$ :

P_2 = P_1 \times \left(\frac{V_1}{V_2}\right) \times \left(\frac{T_2}{T_1}\right)

4. Substitute and solve using dimensional analysis:

P_2 = \pu{1.20 atm} \times \left(\frac{\pu{4.00 L}}{\pu{2.00 L}}\right) \times \left(\frac{\pu{400.15 K}}{\pu{300.15 K}}\right) = \pu{3.20 atm}

Common DAT Exam Trap: Using Celsius temperatures directly in the equation (e.g., using

127.0 / 27.0

instead of

400.15 / 300.15

). This leads to incorrect answers because gas volume and pressure scale directly with absolute thermal energy (Kelvin), not relative Celsius units.

Deriving Classical Gas Laws as Special Cases

By holding one variable constant, the combined gas law simplifies into the specific two-variable laws:

Constant Variable	Resulting Equation	Gas Law Name	Relationship Type
Temperature ( $T_1 = T_2$ )	$P_1 V_1 = P_2 V_2$	Boyle's Law	Inverse (as $V$ decreases, $P$ increases)
Pressure ( $P_1 = P_2$ )	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Charles's Law	Direct (as $T$ increases, $V$ increases)
Volume ( $V_1 = V_2$ )	$\frac{P_1}{T_1} = \frac{P_2}{T_2}$	Gay-Lussac's Law	Direct (as $T$ increases, $P$ increases)

Memory Helper: Keep Peanut Verified on Top! In the combined gas law expression

\frac{PV}{T}

, Pressure and Volume stay in the numerator, while Temperature is always in the denominator.

Critical Thinking Questions

If the absolute temperature of a gas sample doubles while its volume is halved, by what factor does the pressure change?
A rigid steel container holding $\ce{He}$ gas is cooled from $\pu{300.0 K}$ to $\pu{150.0 K}$ . Why does the combined gas law reduce to Gay-Lussac's law in this scenario?
A weather balloon filled with $\ce{He}$ expands as it rises into the stratosphere, where both atmospheric pressure and ambient temperature are lower than at sea level. Explain how decreasing pressure and decreasing temperature exert opposing effects on balloon volume.

References

Sources returned by Google Search grounding during lesson verification.

8.8: The Combined Gas Law - Chemistry LibreTexts — chem.libretexts.org
10.2 Gas Laws Including the Ideal Gas Law - Chad's Prep — chadsprep.com

The Combined Gas Law

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