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Gas Laws – Kelvin Scale, Ideal Gas Law, and Dalton’s Law of Partial Pressures

Absolute Temperature & the Kelvin Scale

  • Only legitimate zero: The Kelvin (absolute) scale is the only scale where 0\,\text{K} coincides with the physical definition of zero temperature (no molecular motion, hypothetically zero volume).
    • Celsius 0\,^{\circ}\text{C} = freezing point of water, not zero motion.
    • Fahrenheit, Rankine, etc. are human-convenience scales; they carry no fundamental meaning at their zero points.
  • Rule for all gas-law work: Temperature must be in Kelvin before substitution.
    • Quick conversion: T{\text{K}} = T{\,^{\circ}\text{C}} + 273 (use 273.15 if significant figures demand it).
  • Exam hint: Data are usually presented in ^{\circ}\text{C}; always convert first to avoid subtle errors.

“Change” (Two-State) Gas-Law Problems

  • When the wording indicates a transition (one pressure/temperature/volume to another), expect variables to appear twice (state 1 → state 2).
  • Combined, Boyle’s, Charles’, Avogadro’s laws, etc. all fall under this umbrella; algebra often involves rearranging a variable in the denominator—practice this.
    • Instructor tip: Rewrite any textbook problem giving P or V as unknown; instead make n or T the unknown and solve to drill the algebra.

Ideal Gas Law Refresher

  • One-state form: PV = nRT
    • P: pressure (atm if you want R=0.0821\,\text{L·atm·mol}^{-1}\text{K}^{-1})
    • V: volume (L)
    • n: moles of gas
    • R: ideal-gas constant (match to chosen units)
    • T: absolute temperature (K)
  • To predict an unknown volume you must know the other three variables exactly; if one is missing (commonly n) you must obtain it by stoichiometry first.

Recognising a Gas Mixture

A problem involves Dalton’s Law if any of the following appear:

  1. The word “mixture” is explicitly used.
  2. Several pure components are listed (e.g. “O$2$, H$2$, N$_2$”).
  3. The sample is described as “air”.
    • Atmospheric air is itself a mixture (major components N$2$, O$2$, plus Ar, CO$2$, H$2$O, etc.).

Dalton’s Law of Partial Pressures

  • Conceptual picture: Different coloured gas particles in one container all bounce off the walls. The wall “feels” only total collisions—cannot distinguish colours.
  • Mathematical statement:
    P{\text{total}} = P1 + P2 + P3 + \dots = \sumi Pi
    where each P_i is a partial pressure for pure component i.

Marrying Dalton with the Ideal Gas Law

  • Same container ⇒ same V for every component and the whole.
  • Same environment ⇒ same T for every component and the whole.
  • R is constant.
  • Therefore, for any component i:
    Pi = \frac{niRT}{V}
    and for the whole sample:
    P{\text{total}} = \frac{n{\text{total}}RT}{V}
  • Only P and n switch between “partial” and “total”.

Worked Example – Partial Pressure of Argon in Air

One-litre sample of air at 25^{\circ}\text{C} and 786\;\text{mmHg} contains 0.00038\;\text{mol Ar}. Find the partial pressure of Ar.

  1. Recognise mixture: “air” → use Dalton.
  2. Identify requested variable: “partial pressure of Ar” → P_{\text{Ar}} unknown.
  3. Set up ideal gas law for that component:
    P{\text{Ar}} = \frac{n{\text{Ar}}RT}{V}
  4. Convert temperature: T = 25 + 273 = 298\,\text{K}
  5. Insert data (units matched to R):
    n{\text{Ar}} = 0.00038\;\text{mol} R = 0.0821\;\text{L·atm·mol}^{-1}\text{K}^{-1} V = 1.00\;\text{L} P{\text{Ar}} = \frac{0.00038\times0.0821\times298}{1.00}
  6. Calculate: P_{\text{Ar}} \approx 0.0093\,\text{atm}
    • Optional conversion: 0.0093\,\text{atm}\times760\,\frac{\text{mmHg}}{\text{atm}} \approx 7.1\,\text{mmHg}
  7. Avoiding common mistake: The given 786\;\text{mmHg} is the total air pressure, not Ar’s partial pressure—do not insert it into the ideal-gas equation for Ar.
  8. Units: Because the question merely says “pressure,” leaving the answer in atm (the natural output of the calculation) is acceptable and avoids an extra conversion step that might introduce error.

Common Pitfalls & Instructor Advice

  • Mixing totals and parts: Always pair n{\text{total}} with P{\text{total}}, ni with Pi. Never cross the two.
  • Temperature oversight: Forgetting to convert to Kelvin invalidates every calculation.
  • Algebra in denominators: Problems that solve for T or n (variables buried in the denominator in Charles’/Avogadro’s/Combined laws) trip students. Systematically practice by “swapping” known/unknown variables in already-solved problems.
  • Over-conversion of units: If the final answer can be presented in the working units (e.g.
    atm from PV=nRT), do so. Only convert if the prompt explicitly demands different units.

Real-World & Conceptual Connections

  • Air composition relevance: Predicting breathing-gas behaviour in scuba tanks, aviation, hyperbaric medicine relies on Dalton’s law.
  • Absolute zero: Foundation for cryogenics, Bose–Einstein condensates, and astrophysical background-temperature studies—practically unreachable but theoretically critical.
  • Ethical dimension: Safety calculations for gas-storage cylinders depend on the precision of partial-pressure predictions; mishandling (misreading total vs partial) can lead to catastrophic failures.

Quick Reference Cheat-Sheet

  • Kelvin conversion: T{\text{K}} = T{\,^{\circ}\text{C}} + 273
  • Ideal gas constant (handy form): R = 0.0821\;\text{L·atm·mol}^{-1}\text{K}^{-1}
  • Dalton’s Law: P{\text{total}} = \sumi P_i
  • Partial pressure via ideal gas law: Pi = \dfrac{niRT}{V}
  • Total moles: n{\text{total}} = \sumi n_i
  • Conversion: 1\,\text{atm} = 760\,\text{mmHg} = 101.3\,\text{kPa}
  • If units are unspecified, stick with the units naturally produced by your chosen form of R.