Note
Studied by 4 people
Ideal Gas Law & Real-Gas Deviations
Ideal Gas Law Fundamentals
Core relationship: PV = nRT
P = pressure of the gas
V = volume it occupies
n = amount of gas in moles
R = universal/ideal‐gas constant (fixed value)
T = absolute temperature (Kelvin)
Purpose: lets us predict one unknown state variable when the other three are measured for one single set of conditions (no “before/after” comparison involved).
Contrasts with combined/“boyle-charles” style gas‐law calculations, which compare an initial state to a final state.
Universal Gas Constant (R)
Adopted in class: R = 0.08206\; \text{atm·L·mol}^{-1}\text{·K}^{-1}
Because R carries units, it dictates the units every other symbol must be in before substituting numbers:
Pressure → \text{atm}
Volume → \text{L} (litres)
Amount → \text{mol}
Temperature → \text{K} (Kelvin)
If any measured quantity is reported in different units, it must be converted.*
Kinetic-Molecular‐Theory (KMT) Backdrop
KMT idealizations
Gas particles have negligible volume relative to the container.
Collisions are perfectly elastic.
No intermolecular attractions/repulsions.
“Ideal” gases perfectly satisfy these assumptions; real gases do not.
Best-case conditions for ideality
Small, light molecules (e.g.
\text{He}
\text{Ne})
High temperature ⇒ kinetic energy overwhelms attractions.
Low pressure ⇒ particles are spaced far apart, so their own volumes & forces matter less.
Real vs. Ideal Gases – Causes of Deviation
Molecular size
Large species (e.g. \text{Xe}, \text{C4H{10}}) occupy non-negligible space; at the same container size they crowd each other, effectively lowering the free volume.
Intermolecular forces
Polar or easily polarised molecules experience attractions (dipole–dipole, dispersion) that reduce impact‐energy, alter collision frequency & pressure.
Example comparison:
\text{He} → very small, non-polar, essentially zero attractions; follows PV=nRT closely.
\text{H_2S} → bent, polar; partial \delta^+ on H attracts \delta^- on S of a neighbour → transient sticking, reduced velocity, lower measured pressure than ideal prediction.
High pressure / Low volume amplifies both factors.
Instructor’s thought experiment: compress xenon into an ever smaller container → predicted vs. real pressure diverges further.
When & Why to Choose PV=nRT
You are given just one snapshot of the system (no "initial vs final" data).
Typical statements: “Gas is measured at ___ psi, occupies ___ mL, contains ___ mg; what is the temperature?”
Starts the workflow:
Collect the four state quantities.
Convert to R-consistent units.
Solve for the unknown algebraically.
Unit-Conversion Toolkit (as used in class)
Pressure
1\;\text{atm} = 14.7\;\text{psi} = 101.3\;\text{kPa} = 760\;\text{torr}
Volume
1000\;\text{mL} = 1\;\text{L}
Mass → moles
n = \dfrac{\text{mass (g)}}{\text{molar mass (g·mol}^{-1})}
Temperature
T\text{(K)} = T{(°C)} + 273.15
Worked-Example Outline (values from transcript)
Given:
• P = 26.2\;\text{psi}
• mass = 97.6\;\text{mg} ("vinegar"—exact formula unspecified)
• V = 854.3\;\text{mL}
• Find T
Step-by-step conversions
Pressure: P_{(\text{atm})} = \dfrac{26.2\;\text{psi}}{14.7\;\text{psi·atm}^{-1}} = 1.78\;\text{atm}
Volume: V_{(\text{L})} = \dfrac{854.3\;\text{mL}}{1000} = 0.8543\;\text{L}
Moles: n = \dfrac{0.0976\;\text{g}}{M} where M = molar mass of the gas (must be supplied or calculated separately).
Rearrange PV=nRT for T:
T = \dfrac{PV}{nR}Substitute:
T = \dfrac{(1.78\;\text{atm})(0.8543\;\text{L})}{\bigl(0.0976\;\text{g}/M\bigr)(0.08206\;\text{atm·L·mol}^{-1}\text{·K}^{-1})}Insert the correct M → compute n → final T (in Kelvin); convert to °C if requested.
Key point: Not a single original number was already in compatible units, so step-zero is always unit surgery.
Density Connection
Rearranging PV = nRT with n = \dfrac{m}{M} gives a handy gas density form:
\rho = \dfrac{m}{V} = \dfrac{PM}{RT}Allows you to predict or confirm gas densities, determine molar mass from measured density, etc.
Frequently printed underneath the standard PV=nRT formula on reference sheets.
Strategy & Problem-Solving Reminders
Identify whether the problem is single-state vs. two-state; that tells you whether to reach for PV=nRT or combined gas law.
Write down required units for R immediately; circle any measurement not already compliant.
If algebra feels messy, numerically plug in after rearranging, or vice-versa—either path is acceptable as long as variables/powers stay clear.
Always keep temperature in Kelvin while the equation is active.
Check your answer’s sanity:
Does the magnitude of T fit the context? (room temp ≈ 298\;\text{K}, boiling water ≈ 373\;\text{K}, etc.)
Very large or small values usually scream “unit error!”
For real gases at high P/low T, expect small systematic deviations (measured P often < ideal P if attractions dominate).
Ethical & Practical Implications
Industrial gas handling (e.g., high-pressure xenon, sulfur gases) must account for deviation to avoid catastrophic under-engineered vessels.
Environmental data logging relies on proper unit conversions to prevent mis-reporting atmospheric concentrations.
Academic integrity tip: show conversions explicitly; partial credit can be rescued even if arithmetic blunders occur.
Connections to Earlier Material
Mirrors the “speed–wavelength” work from wave mechanics: constants impose unit discipline.
Echoes previous stoichiometry chapters: grams → moles is still the bridge between laboratory mass & chemical amount.
Quick Reference Checklist
[ ] All P,V,T,n in correct units
[ ] Identify large vs. small, polar vs. non-polar – anticipate real-gas deviations
[ ] Decide on PV=nRT or combined gas law
[ ] Algebraic isolate the unknown
[ ] Plug & solve, then check plausibility
Ideal Gas Law Fundamentals
The Ideal Gas Law, expressed as PV = nRT, describes the core relationship between the pressure (P), volume (V), amount in moles (n), and absolute temperature (T) of an ideal gas. R represents the universal or ideal-gas constant, a fixed value. This law allows for the prediction of one unknown state variable when the other three are measured under a single set of conditions, unlike combined or "Boyle-Charles" style gas-law calculations which compare an initial state to a final state.
Universal Gas Constant (R)
The universally adopted value for R in this context is 0.08206\;\text{atm\cdot L\cdot mol}^{-1}\text{\cdot K}^{-1}. Due to the units carried by R, all other variables must be in specific units before numerical substitution: pressure in atmospheres (atm), volume in liters (L), amount in moles (mol), and temperature in Kelvin (K). Any measured quantity initially reported in different units must be converted accordingly.
Kinetic-Molecular-Theory (KMT) Backdrop
The Ideal Gas Law is based on the Kinetic-Molecular Theory (KMT), which makes several idealizations about gases: gas particles have negligible volume relative to the container, collisions between particles are perfectly elastic, and there are no intermolecular attractions or repulsions. While "ideal" gases perfectly satisfy these assumptions, real gases do not. The best-case conditions for a real gas to behave ideally include having small, light molecules (e.g., He, Ne), high temperature (where kinetic energy overwhelms intermolecular attractions), and low pressure (where particles are spaced far apart, minimizing the impact of their own volumes and forces).
Real vs. Ideal Gases – Causes of Deviation
Deviations of real gases from ideal behavior are primarily caused by two factors: molecular size and intermolecular forces. Large species (e.g., Xe, C4H{10}) occupy non-negligible space, which effectively lowers the free volume available to other particles within the same container, leading to crowding. Additionally, polar or easily polarizable molecules experience intermolecular attractions (such as dipole-dipole or dispersion forces) that reduce impact energy and alter collision frequency, consequently affecting the measured pressure. For example, He, being very small and non-polar with essentially zero attractions, follows PV=nRT closely. In contrast, H_2S, which is bent and polar, exhibits partial positive charges on hydrogen attracting negative charges on sulfur of neighboring molecules, leading to transient sticking, reduced velocity, and a lower measured pressure than ideal predictions. High pressure or low volume conditions amplify both these factors, causing a greater divergence between predicted and real pressure, as demonstrated by compressing xenon into a progressively smaller container.
When & Why to Choose PV=nRT
PV=nRT is the appropriate formula when you are given just one snapshot of the system, meaning there is no "initial vs. final" data to compare. Typical problem statements might ask for temperature when pressure, volume, and mass are known (e.g., “Gas is measured at ___ psi, occupies ___ mL, contains ___ mg; what is the temperature?”). The workflow begins by collecting the four state quantities, converting them to R-consistent units, and then algebraically solving for the unknown.
Unit-Conversion Toolkit
Essential unit conversions for using the Ideal Gas Law are:
Pressure
1\;\text{atm} = 14.7\;\text{psi} = 101.3\;\text{kPa} = 760\;\text{torr}.
Volume
1000\;\text{mL} = 1\;\text{L}.
Mass to Moles
Amount in moles (n) is calculated as n = \dfrac{\text{mass (g)}}{\text{molar mass (g\cdot mol}^{-1})}.
Temperature
Temperature in Kelvin (T\text{(K)}) is obtained by adding 273.15 to the temperature in Celsius (T\text{(°C)}): T\text{(K)} = T\text{(°C)} + 273.15.
Worked-Example Outline
Consider obtaining T given P = 26.2\;\text{psi}, mass of "vinegar" = 97.6\;\text{mg}, and V = 854.3\;\text{mL}. The step-by-step conversions are crucial:
Pressure: Convert psi to atm: P_{\text{(atm)}} = \dfrac{26.2\;\text{psi}}{14.7\;\text{psi\cdot atm}^{-1}} = 1.78\;\text{atm}.
Volume: Convert mL to L: V_{\text{(L)}} = \dfrac{854.3\;\text{mL}}{1000} = 0.8543\;\text{L}.
Moles: Calculate moles (n) using the mass and molar mass (M), where n = \dfrac{0.0976\;\text{g}}{M}. The molar mass (M) must be supplied or determined separately.
Rearrange: Algebraically solve PV=nRT for T: T = \dfrac{PV}{nR}.
Substitute: Plug in the converted values: T = \dfrac{(1.78\;\text{atm})(0.8543\;\text{L})}{\bigl(0.0976\;\text{g}/M\bigr)(0.08206\;\text{atm\cdot L\cdot mol}^{-1}\text{\cdot K}^{-1})}.
Compute: Insert the correct M to compute n, then calculate the final T in Kelvin. If requested, convert to °C. This example highlights that typically not a single original number is in compatible units, making unit conversion the indispensable "step zero."
Density Connection
By rearranging PV = nRT and substituting n = \dfrac{m}{M}, a useful gas density form is derived: \rho = \dfrac{m}{V} = \dfrac{PM}{RT}. This equation enables the prediction or confirmation of gas densities and the determination of molar mass from measured density. It is frequently presented alongside the standard Ideal Gas Law on reference sheets.
Strategy & Problem-Solving Reminders
When approaching gas law problems, first identify whether it's a single-state or two-state problem to determine if PV=nRT or the combined gas law is needed. Immediately write down the required units for R and circle any measurements that are not compliant. If algebra feels complex, either numerically plug in values after rearranging or vice-versa, ensuring variables and powers remain clear. Always keep temperature in Kelvin throughout the equation. Finally, check your answer's sanity: Does the magnitude of T fit the context (e.g., room temp \approx 298\;\text{K}, boiling water \approx 373\;\text{K})? Very large or small values often indicate a unit error. For real gases at high pressure or low temperature, expect small systematic deviations, with measured pressure often being less than the ideal pressure if attractions are dominant.
Ethical & Practical Implications
Understanding gas behavior is critical for industrial gas handling, especially with high-pressure gases like xenon or sulfur gases, to prevent catastrophic under-engineered vessels due to deviations from ideal behavior. Environmental data logging also relies on proper unit conversions to accurately report atmospheric concentrations. Academically, explicitly showing conversions can rescue partial credit even if arithmetic errors occur.
Connections to Earlier Material
The principles discussed here mirror the "speed-wavelength" work from wave mechanics, where constants impose unit discipline. It also echoes previous stoichiometry chapters, where grams-to-moles conversion remains the bridge between laboratory mass and chemical amount.
Quick Reference Checklist
To ensure problem-solving success:
[ ] All P,V,T,n in correct units
[ ] Identify large vs. small, polar vs. non-polar – anticipate real-gas deviations
[ ] Decide on PV=nRT or combined gas law
[ ] Algebraically isolate the unknown
[ ] Plug & solve, then check plausibility