Gas Laws — Gay-Lussac, Charles & Boyle

Fundamental Gas Variables

• When studying gases we track four core state variables:

  • Pressure, $P$, typically measured in atmospheres (atm) in this course, but other units (kPa, Torr) appear in examples.

  • Volume, $V$, expressed in liters (L).

  • Temperature, $T$, always converted to Kelvin (K) for calculations.

  • Amount of substance, $n$, measured in moles (mol).

• In every gas-law problem, identify which of these variables change and which stay constant.

• Converting to Kelvin: $T<em>{(K)} = T</em>{(^\circ C)} + 273.15$. This keeps all temperatures on an absolute scale so direct and inverse proportionalities work algebraically.

The Ideal-Gas Constant, $R$

• Introduced only by name in this segment; numerical values (e.g. $0.0821\;\text{L·atm·mol}^{-1}\text{K}^{-1}$) come later when the Ideal-Gas Equation $PV = nRT$ is derived.

• Key point: all four simple gas laws discussed here will later “collapse” into the ideal-gas equation once $R$ is introduced.

Overview of the Four Simple Gas Laws

• Each law holds two variables constant and relates the other two:

  1. Gay-Lussac’s Law – $P$ vs. $T$ (constant $V$, $n$).

  2. Charles’s Law – $V$ vs. $T$ (constant $P$, $n$).

  3. Boyle’s Law – $P$ vs. $V$ (constant $T$, $n$).

  4. (Mentioned but not yet covered in the clip) Avogadro’s or a variant relating $V$ and $n$ at constant $P, T$.

Gay-Lussac’s Law (Pressure–Temperature Relationship)

• Qualitative statement: At constant volume and amount, the pressure of a gas is directly proportional to its absolute temperature.

• Mathematical form used in lecture:
  $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2} = k \qquad(\text{with }V,n\text{ constant})$

• The transcript shows a common typo ("$P<em>1V</em>1 = P<em>2V</em>2$"). Instructor intent is clearly $P<em>1/T</em>1 = P<em>2/T</em>2$ for Gay-Lussac.

• Graphical idea: Straight line through origin on a $P$–$T$ plot (Kelvin).

Worked Example 1 – Propellant in a Hairspray Can

• Scenario: Can labelled “Do not incinerate. May burst above $120\;^\circ\text{F}$.”

• Data listed:

  • $P_1 = 360\;\text{kPa}$

  • $T_1 = 24\;^\circ\text{C} = 297.15\;\text{K}$

  • $T_2 = 50\;^\circ\text{C} = 323.15\;\text{K}$

  • $V,n$ unchanged (the can is sealed). Solve for $P_2$.

• Calculation:
  $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2} \;\Rightarrow\; \frac{360}{297.15} = \frac{P<em>2}{323.15}$ $P</em>2 = 391\;\text{kPa}$ (rounded in lecture; keep units!).

• Conceptual takeaway: Heating a sealed can increases pressure—eventually enough to rupture the container (real-world safety relevance).

Worked Example 2 – Cooling Nitrogen Gas

• Data:

  • $V = 45\;\text{mL}$ (constant)

  • $P_1 = 600\;\text{Torr}$

  • $T_1 = 27\;^\circ\text{C} = 300\;\text{K}$

  • $T_2 = -73\;^\circ\text{C} = 200\;\text{K}$

• Setup:
  $\frac{600}{300} = \frac{P<em>2}{200} \;\Rightarrow\; P</em>2 = 400\;\text{Torr}$

• Quick-ratio reasoning offered: $T<em>2/T</em>1 = \tfrac{2}{3}$ so $P<em>2 = \tfrac{2}{3}P</em>1$.

Conceptual Shortcuts for Direct Proportions

• If $P \propto T$ or $V \propto T$, any fractional change in one variable produces the same fractional change in the other.

• Instructor encourages mental checks but still wants full algebra shown in formal work.

Charles’s Law (Volume–Temperature Relationship)

• Statement: At constant pressure and amount, volume is directly proportional to absolute temperature.

• Formula:
  $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$

• Example Prompt (CO$_2$):

  • Given: $V<em>1 = 0.300\;\text{L},\;T</em>1 = 10\;^\circ\text{C}$, $P = 750\;\text{Torr}$ (constant).

  • Find $V<em>2$ at $T</em>2 = 30\;^\circ\text{C}$.

  • Students instructed to convert to Kelvin, plug into the formula, and report the answer in liters (same unit supplied).

• Additional practice problems follow the same template; rule of thumb reiterated: the unit of your final answer should match the unit initially provided, even if intermediate conversions were needed.

Boyle’s Law (Pressure–Volume Relationship)

• Statement: At constant temperature and amount, pressure of a gas is inversely proportional to its volume.

• Algebraic form:
  $P<em>1V</em>1 = P<em>2V</em>2$

• Conceptual example provided (no numbers): “If the volume is cut in half, the pressure must double.”

• Inverse relationship means a hyperbolic curve on a $P$–$V$ plot.

General Problem-Solving Strategy Highlighted by Instructor

• 1 – List given and unknown quantities with units (e.g. $P<em>1, T</em>1, P<em>2, T</em>2$).

• 2 – Convert temperatures to Kelvin; convert other units only when mixing incompatible units (kPa vs Torr vs atm).

• 3 – Select the correct law (look at which variables are stated as “the same” or “unchanged”).

• 4 – Insert data into the proportionality or cross-multiplication form.

• 5 – Solve algebraically and include units in every step.

• 6 – Perform a quick reasonableness check (direct vs inverse trends, fraction logic, significant-figure sanity).

Unit Conversions & Notation Tips

• Pressure:

  • $1\;\text{atm} = 760\;\text{Torr} = 101.325\;\text{kPa}$ (useful when you must change units to match $R$ in the ideal-gas equation later).

• Temperature: add $273.15$ to go from $^\circ\text{C}$ to K; subtract to reverse.

• Volume: $1000\;\text{mL} = 1\;\text{L}$. Keep eye on mL vs L; Charles’s & Boyle’s laws work with any consistent volume unit as long as the ratio is preserved.

Connections to Broader Topics

• These simple laws stem from the Kinetic Molecular Theory (discussed in earlier lectures): temperature reflects average kinetic energy; pressure arises from molecular collisions; changing volume alters collision frequency.

• Ethical / safety implication: Warnings on aerosol cans are grounded in Gay-Lussac’s Law—heating a sealed container raises internal pressure and can cause dangerous ruptures.

• Mathematically, combining the three empirical laws gives the Combined Gas Law $\frac{P<em>1V</em>1}{T<em>1}=\frac{P</em>2V<em>2}{T</em>2}$, which leads directly to the Ideal-Gas Equation once $nR$ is introduced.

These notes encapsulate every numerical example, formula, shortcut, and conceptual remark provided in the transcript, formatted for quick review before an exam.