Gas Laws – Review Problems & Test Preparation

Review Problem #1 – Mixing Three Separate Gas Samples

• Scenario: Three rigid containers (each previously sealed) are connected via a volumeless valve; once the valve is opened, the gases freely inter-diffuse and occupy the combined space.

• Given data before mixing
– Container A (N₂): $P{N2,\;initial}=265\;\text{Torr},\;V{A}=1.0\;\text{L}$ – Container B (Ne): $P{Ne,\;initial}=800\;\text{Torr},\;V{B}=1.0\;\text{L}$ – Container C (H₂): $P{H2,\;initial}=532\;\text{Torr},\;V{C}=0.50\;\text{L}$
→ Total final volume after the valve is opened: $V_{total}=1.0+1.0+0.50=2.5\;\text{L}$
(Temperature and amount of each gas are unchanged; the only variable that changes is volume.)

• Physics/chemistry principle: Boyle’s Law $P1V1=P2V2$ for each gas separately, followed by Dalton’s Law of Partial Pressures $P{tot}=\sum Pi$ when all partial pressures refer to the same, common volume.

• Step-wise calculation (shown for each gas):

Solve for new partial pressure $P{i,\;final}=P{i,\;initial}\left(\dfrac{V{i,\;initial}}{V{total}}\right)$
Compute numerical values:
– N₂ : $P{N2}=265\;\text{Torr}\times\dfrac{1.0}{2.5}=106\;\text{Torr}$
– Ne : $P{Ne}=800\;\text{Torr}\times\dfrac{1.0}{2.5}=320\;\text{Torr}$ – H₂ : $P{H_2}=532\;\text{Torr}\times\dfrac{0.50}{2.5}=106\;\text{Torr}\;(106.4\;\text{Torr rounded})$

• Total pressure after mixing:
$P_{tot}=106+320+106 \;\text{Torr}=532\;\text{Torr}$

• Conceptual takeaways
– Each gas obeys Boyle’s Law independently because $n$ and $T$ are constant for that gas.
– Dalton’s Law assumes gas particles do not interact (ideal behaviour). Once the gases share a volume, each exerts its own pressure proportional to its mole fraction and the common $T,V$ .

Review Problem #2 – Gas Collected Over Water (Dalton’s Law + Combined Gas Law)

• Lab technique: A gas is captured by water displacement. The collected volume actually contains dry gas + water vapour. Measured pressure is the sum of the two.

• Given data
– Manometer reading: $P{meas}=103.64\;\text{kPa}$ – Water-vapour pressure at the bath temperature: $P{H2O}=21.60\;\text{kPa}$ – Collected volume: $V1=500\;\text{mL}=0.500\;\text{L}$
– Bath temperature: $T1=292\;\text{K}$ – Target conditions: STP $(P2=1\;\text{atm}=101.325\;\text{kPa},\;T_2=273\;\text{K})$

• Step 1 – Isolate the pressure of the dry gas (Dalton):
$P{dry}=P{meas}-P{H2O}=103.64-21.60=82.04\;\text{kPa}$

• Step 2 – Use the Combined Gas Law to convert to STP volume:
$\frac{P1V1}{T1}=\frac{P2V2}{T2}\quad\Rightarrow\quad V2=\frac{P1V1T2}{P2T1}$
$V_2=\frac{82.04\;\text{kPa}\times0.500\;\text{L}\times273\;\text{K}}{101.325\;\text{kPa}\times292\;\text{K}}=0.378\;\text{L}=378\;\text{mL}$

• Key facts & reminders
– Always subtract the vapour pressure of water to obtain the pressure of the dry sample.
– STP is 0 °C (273 K) and 1 atm (101.325 kPa).
– Temperatures must be in Kelvin for gas-law calculations.

Review Problem #3 – Initial–Final State with Constant V & T (Gay-Lussac-Type)

• Scenario: Rigid 500 mL vessel at constant room temperature $T=295\;\text{K}$ initially holds 100 g of CO₂. An additional 50 g CO₂ is injected. Find the new pressure.

• Quantities that stay constant: $V,\;T$
Quantity changing: $n$ (moles) → pressure will scale with $n$ .

• Relationship (Gay-Lussac variant): $\displaystyle\frac{P1}{n1}=\frac{P2}{n2}\quad\Rightarrow\quad P2=P1\left(\frac{n2}{n1}\right)$

• Convert masses to moles using molar mass $M{CO2}=44.007\;\text{g\,mol}^{-1}$
– Initial: $n1=\frac{100}{44.007}=2.272\;\text{mol}$ – Final: $n2=\frac{150}{44.007}=3.408\;\text{mol}$

• Compute new pressure:
$P_2=765\;\text{mmHg}\times\frac{3.408}{2.272}=1.148\times10^{3}\;\text{mmHg}\;(1.148\;\text{atm}\times760\;\text{mmHg/atm})$

• Practical considerations
– Rigid, constant-temperature container ⇒ only $P$ scales with $n$ .
– If any variable other than $P$ or $n$ were allowed to change, the ideal gas law or another combined relationship would be required.

Exam & Conceptual Guide (Chs 18 & 19)

1. Standard Temperature and Pressure (STP)

• Definition: $T=273\;\text{K}$ (0 °C), $P=1\;\text{atm}=101.325\;\text{kPa}=760\;\text{Torr}=760\;\text{mmHg}$
• Questions may simply say “STP”; know both the concept and the numerical values.

2. Ideal vs. Real Gases

• Ideal-gas postulates:
– Point-like particles with negligible volume.
– No intermolecular forces; collisions are perfectly elastic.
– Gas obeys $PV=nRT$ at all $T,P,n,V$ .

• Real gases deviate at:
– High pressure: particles are forced close together → finite molecular volume + significant Coulombic attractions (reduced $P$ ).
– Low temperature: kinetic energy $\left(\tfrac12 m v^2\right)$ is insufficient to overcome attractions → trajectory deviates, effective $P$ drops.

3. Graphical & Mathematical Relationships

• Direct vs inverse proportionalities (e.g., Boyle, Charles, Gay-Lussac).
• Be able to recognize or sketch linear vs hyperbolic plots.

4. Units & Conversions

• Common pressure units: $\text{atm},\;\text{kPa},\;\text{Torr},\;\text{mmHg}$ .
• Always convert Celsius → Kelvin before calculation.
• Occasional problems may embed mixed units; be ready.

5. Diffusion & Graham’s Law

• Concept: rate of diffusion/effusion $\propto\dfrac{1}{\sqrt{M}}$ (
$\dfrac{v1}{v2}=\sqrt{\dfrac{M2}{M1}}$ ).
• Two common tasks:

“How many times faster/slower?” (velocity ratio).
“How long will it take?” (time inversely proportional to velocity).

6. Dalton’s Law Variants

• Total vs partial pressures in a mixture.
• Gas collected over water (subtract vapour pressure).
• Partial-pressure adjustment when volume changes (as in Review Problem #1).

7. “Initial → Final” State Problems

• Identify which variables change; apply the appropriate simplified law or the full combined gas law.
• Examples: constant $V,T$ (Gay-Lussac-type), constant $P,T$ (Charles-type), constant $P,V$ (Avogadro-type), etc.

8. Allowed Resources

• You may use a personal formula sheet during the test.
• Extensive practice worksheets + answer keys are posted (check Edmodo).

9. Practical Study Tips

• Work through assorted practice problems to build speed.
• Focus on unit consistency; many errors are unit-related.
• Understand why each law applies, not just which plugged-in numbers produce an answer.
• Email the instructor for clarification as needed.

Ethical & Real-World Connections

• Gas-law deviations underpin industrial liquefaction (e.g., LNG plants), high-pressure scuba considerations, and atmospheric science.
• Understanding vapour-pressure corrections is vital in environmental monitoring where humidity skews sensor readings.
• Awareness of ideal-vs-real behaviour prevents unsafe assumptions—e.g., cylinder filling under high pressure.

Formula Summary (Quick Reference)

• Ideal gas law: $PV=nRT$
• Boyle’s: $P1V1=P2V2$ (constant $n,T$ )
• Charles’s: $\dfrac{V1}{T1}=\dfrac{V2}{T2}$ (constant $n,P$ )
• Gay-Lussac’s: $\dfrac{P1}{T1}=\dfrac{P2}{T2}$ (constant $n,V$ )
• Combined: $\dfrac{P1V1}{T1}=\dfrac{P2V2}{T2}$
• Dalton’s: $P{tot}=\sumi Pi$ • Graham’s: $\dfrac{v1}{v2}=\sqrt{\dfrac{M2}{M1}}$ • Mole fraction: $Xi=\dfrac{ni}{n{tot}},\;Pi=XiP_{tot}$

Use the reference sheet during practice to ensure fluency with variable isolation and unit handling.