Note
Studied by 4 people
Gas Pressure, Kinetic-Molecular Theory & Combined Gas Law
Pressure – Core Definition & Intuition
Physical definition: P = \frac{F}{A}
F = force (push or pull, produced by mass ⋅ gravity, collisions, etc.)
A = contact area over which that force is distributed.
Relationships
Direct with force → ↑ force at same area → ↑ pressure.
Inverse with area → same force over ↓ area → ↑ pressure.
Everyday illustrations
“Bed-of-nails” demonstration
Many nails = large total area → body weight spread out → pressure on each nail is low → skin not punctured.
Stepping on a single nail = very small area → same body weight focused → high pressure → puncture & need for tetanus shot.
Exact same principle governs gas pressure inside containers.
Atmospheric & Weather Context
Atmospheric pressure = weight of air molecules colliding with Earth’s surface.
High-pressure system: denser parcel of air, particles moving downward, exerting more collisions → people sensitive to pressure feel it (joint aches, storm “pressure headaches”).
Low-pressure system: air rising/ diverging, fewer downward collisions → often associated with storms, snow, rain.
Kinetic-Molecular Theory (KMT) – 5 Postulates for an Ideal Gas
Continuous random motion
Particles travel in straight lines at assorted speeds until they collide with another particle or with container walls, then instantly change direction.
Negligible particle volume
Actual molecular size ≪ average inter-particle distance → container is mostly empty space; therefore the gas’s “volume” = volume of the container.
Pressure arises solely from wall collisions
Particle-particle collisions change directions but do not create macroscopic pressure; only particle–wall impacts transfer momentum that we register as pressure.
No intermolecular attractions or repulsions (ideal assumption)
Molecules behave like hard spheres that neither stick together nor repel; therefore collisions are perfectly elastic → no kinetic-energy loss.
Average kinetic energy ∝ absolute temperature (Kelvin)
\overline{KE} = \frac{3}{2}kB T (or (\frac{1}{2}m v{\text{rms}}^{2}))
Same temperature ⇒ all gases share the same average kinetic energy, even though individual speeds differ.
Maxwell–Boltzmann Distributions
Graph: number of molecules vs. speed.
273 K curve: peak at a moderate speed; long tail for faster molecules; some very slow ones present.
500 K curve: shifted right (higher most-probable speed) and flattened (broader spread); still some slow particles but more very fast ones.
Mass effect via KE = \frac{1}{2} m v^{2}
At identical T, heavier atoms (Xe, Kr) have greater m so their root-mean-square speed v_{\text{rms}} is lower yet KE is identical to lighter gases (He, Ne).
Real-world: semitruck vs. smart-car collision analogy—large-mass truck hard to stop because KE escalates with mass.
Fundamental Gas-Variable Relationships
Variables: Pressure (P), Volume (V), Temperature (T in K), Amount (n in moles).
1. Pressure–Temperature (Gay-Lussac’s Law)
Hold n and V constant (rigid container).
Simulation observation: heating speeds up particles → more frequent/harder wall hits → ↑P; cooling slows them → ↓P.
Mathematical form: \frac{P1}{T1} = \frac{P2}{T2} (direct proportionality).
Doubling T (in K) exactly doubles P.
2. Volume–Temperature (Charles’s Law)
Hold n and P constant (expandable piston).
Heating → expansion; cooling → contraction.
Equation: \frac{V1}{T1} = \frac{V2}{T2} (direct proportionality).
3. Combined Gas Law (all variables mobile)
Generalised expression: \frac{P1 V1}{n1 T1} = \frac{P2 V2}{n2 T2}.
Derives from merging Boyle, Charles, and Gay-Lussac.
Drop any factor that is explicitly held constant to simplify problem solving.
Practical advice
Identify variables by their units (mmHg → pressure, L/mL → volume, °C/K → temperature, mol → amount).
Convert temperatures to Kelvin: T(K) = T(°C) + 273.15.
Make sure matching units appear on both sides (e.g.
ATM with ATM, or mmHg with mmHg.
L with L or convert mL → L).
Solve via cross-multiply & divide; keep significant figures.
Worked Example (Textbook Q16-14)
"A sample of ideal gas occupies 298 mL at 377 mmHg. What volume will it occupy at 522 mmHg if temperature is unchanged?"
Variables
P1 = 377\,\text{mmHg}, \; V1 = 298\,\text{mL}
P2 = 522\,\text{mmHg}, T1 = T_2 \;(\text{constant}), n \text{ constant}
Use reduced relationship P1 V1 = P2 V2.
Solve: V2 = \frac{P1 V1}{P2} = \frac{377 \times 298}{522}\,\text{mL} \approx 2.15 \times 10^{2}\,\text{mL} (≈ 215 mL).
Elastic vs. Inelastic Collisions – Energy Perspective
Car crash analogy: real cars crumple (inelastic) → kinetic energy converts to heat/sound; vehicles feel hot after a collision.
Gas particles under KMT: perfectly elastic → no KE lost to heat, so average T remains unchanged by internal collisions.
Key Take-Home Points
Pressure is fundamentally a contact phenomenon (force over area).
Kinetic-molecular theory provides the microscopic rationale for macroscopic gas laws.
Temperature in Kelvin acts as a direct gauge of average molecular kinetic energy.
All simple gas laws are subsets of the combined gas law; cancel constants to streamline calculations.
Consistent units & Kelvin temperatures are mandatory for valid answers.
Pressure – Core Definition & Intuition
Pressure is defined physically as force (F) distributed over a contact area (A), represented by the formula P = \frac{F}{A}. Force is a push or pull, generated by factors such as mass times gravity or collisions. Pressure has a direct relationship with force: increasing the force over the same area results in increased pressure. Conversely, pressure is inversely proportional to area: applying the same force over a smaller area leads to higher pressure. This concept can be illustrated through everyday examples like a "bed-of-nails" demonstration. When lying on many nails, the large total contact area distributes body weight widely, resulting in low pressure on each individual nail, which prevents skin puncture. However, stepping on a single nail concentrates the same body weight over a very small area, creating high pressure that can cause a puncture. This exact principle also governs gas pressure inside containers.
Atmospheric & Weather Context
Atmospheric pressure is the weight of air molecules colliding with the Earth’s surface. A high-pressure system indicates a denser parcel of air where particles are moving downward, exerting more collisions. Individuals sensitive to pressure often experience symptoms like joint aches or "storm pressure headaches" due to these systems. Conversely, a low-pressure system involves air rising or diverging, leading to fewer downward collisions, and is frequently associated with adverse weather conditions such as storms, snow, or rain.
Kinetic-Molecular Theory (KMT) – 5 Postulates for an Ideal Gas
The Kinetic-Molecular Theory (KMT) for an ideal gas is based on five postulates. First, particles are in continuous random motion, traveling in straight lines at various speeds, changing direction only upon collision with another particle or container walls. Second, the actual volume of gas particles is negligible compared to the average distance between them, meaning the container is mostly empty space, and thus the gas's volume is effectively the volume of the container itself. Third, pressure is generated exclusively by collisions between particles and the container walls; particle-particle collisions only alter direction and do not contribute to macroscopic pressure. Fourth, ideal gas molecules are assumed to have no intermolecular attractions or repulsions, behaving as hard spheres that do not stick together or repel, ensuring perfectly elastic collisions where no kinetic energy is lost. Finally, the average kinetic energy of gas particles is directly proportional to the absolute temperature in Kelvin, expressed as \overline{KE} = \frac{3}{2}k*B T (or \frac{1}{2}m v_{\text{rms}}^{2}). This implies that at the same temperature, all gases share the same average kinetic energy, despite individual particle speeds varying.
Maxwell–Boltzmann Distributions
Maxwell–Boltzmann Distributions are typically plotted as the number of molecules versus their speed. For instance, a 273 K curve shows a peak at a moderate speed, with a long tail indicating faster molecules and the presence of some very slow ones. A 500 K curve, however, is shifted to the right, signifying a higher most-probable speed, and becomes flattened, indicating a broader spread of speeds, with more very fast particles although some slow ones are still present. This distribution also illustrates the mass effect through the kinetic energy formula KE = \frac{1}{2} m v^{2}. At an identical temperature, heavier atoms like Xenon (Xe) or Krypton (Kr) have a greater mass (m), resulting in a lower root-mean-square speed (v_{\text{rms}}) even though their kinetic energy remains identical to lighter gases such as Helium (He) or Neon (Ne). This concept can be compared to a real-world analogy of a semitruck versus a smart car collision: a large-mass truck is harder to stop because its kinetic energy escalates significantly with mass.
Fundamental Gas-Variable Relationships
Variables: Pressure (P), Volume (V), Temperature (T in K), Amount (n in moles).
1. Pressure–Temperature (Gay-Lussac’s Law)
When the amount of gas (n) and volume (V) are held constant, such as in a rigid container, heating the gas increases particle speed, leading to more frequent and harder collisions with the container walls, thus increasing pressure (P). Conversely, cooling slows particles, leading to a decrease in pressure. Mathematically, this direct proportionality is expressed as \frac{P1}{T1} = \frac{P2}{T2}, meaning that doubling the temperature in Kelvin precisely doubles the pressure.
2. Volume–Temperature (Charles’s Law)
When the amount of gas (n) and pressure (P) are held constant, such as in an expandable piston, heating the gas causes it to expand, while cooling leads to contraction. This direct proportionality is described by the equation \frac{V1}{T1} = \frac{V2}{T2} .
3. Combined Gas Law (all variables mobile)
The Combined Gas Law provides a generalized expression: \frac{P1 V1}{n1 T1} = \frac{P2 V2}{n2 T2}. This law is derived from merging Boyle's, Charles's, and Gay-Lussac's laws. For problem-solving, any factor that is explicitly held constant can be dropped from the equation for simplification. Practical advice for using this law includes: first, identifying variables by their units (e.g., mmHg for pressure, L/mL for volume, °C/K for temperature, mol for amount); second, always converting temperatures to Kelvin using the formula T(K) = T(°C) + 273.15; third, ensuring matching units appear on both sides of the equation (e.g., ATM with ATM, or mmHg with mmHg, and L with L or converting mL to L); and finally, solving via cross-multiply and divide while maintaining appropriate significant figures.
Worked Example (Textbook Q16-14)
Consider the problem: "A sample of ideal gas occupies 298 mL at 377 mmHg. What volume will it occupy at 522 mmHg if temperature is unchanged?" The given variables are P1 = 377\,\text{mmHg} and V1 = 298\,\text{mL}, with P2 = 522\,\text{mmHg}. Since both temperature (T1 = T2) and the amount of gas (n) are constant, the reduced relationship from the Combined Gas Law, Boyle's Law (P1 V1 = P2 V2), is used. To solve for V2, rearrange the equation to V2 = \frac{P1 V1}{P2}, which gives V*2 = \frac{377 \times 298}{522}\,\text{mL} \approx 2.15 \times 10^{2}\,\text{mL} (approximately 215 mL).
Elastic vs. Inelastic Collisions – Energy Perspective
Understanding collisions from an energy perspective distinguishes between elastic and inelastic types. An analogy using car crashes illustrates inelastic collisions: real cars crumple, converting kinetic energy into heat and sound, which is why vehicles feel hot after an impact. In contrast, gas particles described by the Kinetic-Molecular Theory (KMT) undergo perfectly elastic collisions, meaning no kinetic energy is lost as heat; consequently, the average temperature of the system remains unchanged by these internal collisions.
Key Take-Home Points
In summary, pressure is fundamentally a contact phenomenon, defined as force distributed over an area. The Kinetic-Molecular Theory offers the microscopic basis for understanding macroscopic gas laws. Temperature, when expressed in Kelvin, serves as a direct measure of the average molecular kinetic energy. All simpler gas laws are merely specific cases of the Combined Gas Law, and constants can be cancelled to simplify calculations. Finally, consistent units and temperatures expressed in Kelvin are essential for obtaining valid results in gas law calculations.