2.4.2 · D5States of Matter (Quantitative)
Question bank — Combined gas law and ideal gas equation PV = nRT
True or false — justify
Decide true/false, then defend it in one sentence.
If you double the Kelvin temperature of a sealed rigid tank, the pressure doubles
True — rigid means fixed and sealed means fixed, so ; doubling (in kelvin) doubles .
Doubling the temperature from to doubles the pressure of a sealed rigid gas
False — you must use kelvin; is only a rise, not a doubling, because absolute zero is at , not .
has different numerical values ( vs ) because it is a different physical constant in each
False — it is the same constant; only the units change, so the number changes with them (Pa·m³ vs atm·L).
At the same temperature and pressure, mol of and mol of occupy the same volume
True — Avogadro's Law and depend only on , not on what the molecules are; equal forces equal .
At the same temperature and pressure, g of and g of occupy the same volume
False — equal mass means very different moles (), and volume tracks moles, so the lighter (smaller ) occupies far more.
The combined gas law is a separate law that must be memorised on its own
False — it is just with constant for a sealed sample; is the same on both sides so it cancels.
Gas density falls as you compress it into a smaller volume
False — : compression raises , so at fixed density rises; more mass in less space.
Two different gases at the same and have the same density
False — depends on molar mass ; the heavier gas is denser even at identical .
The molar volume of any ideal gas at STP is about regardless of its identity
True — at , fixed give one volume via , and never enters, so identity is irrelevant.
Spot the error
Each statement contains one flaw — name it.
"A student writes in for a room-temperature gas."
The error is Celsius; must be absolute, so — using underestimates twelve-fold.
"Pressure is , so I'll use ."
Unit mismatch — pairs with atm and litres; with kPa you must either convert to atm or switch to SI (, Pa, m³).
"The gas leaked, but the sample is still one substance, so I use ."
The combined law needs constant ; a leak changes , so you must return to full for each state separately.
", therefore — I just add the effects."
Separate proportionalities multiply, not add; each variable scales by a factor, and independent factors combine as a product.
"Since , at constant pressure and volume are directly proportional."
They are inversely proportional (Boyle) — is the constant, so as one rises the other falls, not rises together.
"I found , so I halved it because there are two atoms."
from is already the full molar mass of the molecule (); no halving — the equation counts moles of molecules, not atoms.
"At the ideal gas equation predicts zero volume, which is physically exact."
The equation predicts , but real gases liquefy long before that, so this is an extrapolation, not a real observable state.
Why questions
Answer with the reasoning, not just a fact.
Why must temperature be in kelvin and never Celsius in ?
The laws come from measured from absolute zero; Celsius has an arbitrary zero (freezing water), so ratios like would be wrong.
Why does vanish from the combined gas law?
For a sealed sample and are both fixed, so is the same constant in every state and cancels between the two sides.
Why can we multiply the three proportionalities together to get ?
Each law tells how responds to one variable with the others held fixed; independent responses combine multiplicatively, giving the joint dependence.
Why is called the universal gas constant?
Because the same number works for every ideal gas — the whole point is that gas identity drops out, leaving one constant for all.
Why does gas density rise with molar mass at fixed ?
From , at fixed the same number of molecules occupies the same volume, so heavier molecules pack more mass in — density scales with .
Why is tied to STP and not just any conditions?
Molar volume depends on and ; only at the specified STP values does it evaluate to .
Why does compressing a gas and heating it both raise the pressure?
Smaller volume makes collisions with the walls more frequent; higher temperature makes molecules faster and hit harder — both push up (see the kinetic-theory figure above, and Kinetic Theory of Gases).
Why does the ideal gas equation eventually fail at high pressure or low temperature?
It assumes point particles with no attractions; when packed close or moving slowly, molecular volume and attractions matter — corrected by the Real Gases and van der Waals Equation.
Edge cases
Boundary and degenerate scenarios — where careless rules break.
The equation can reach two different ways — which physical routes do and correspond to?
At fixed , cooling to drives (volume shrinks); at fixed , cooling drives (molecules stop hitting walls) — the same limit, but one collapses the box and the other kills the pushing.
What does predict as at fixed , and why can a real gas never actually reach it?
It predicts , but molecules have real size and attract one another, so the gas condenses to a liquid then freezes to a solid — both occupy finite volume — long before zero (see the shrinking-then-condensing figure below).
Can be used in ?
With no gas () the equation gives — consistent (a true vacuum exerts no pressure), but the combined law divides by nothing meaningful, so it simply does not apply.
In a mixture of two gases, does use the total moles or moles of one gas?
Total gives total pressure; each component obeys with its own partial pressure — that is Dalton's Law of Partial Pressures (partial-pressure figure above).
If a rigid sealed tank is heated until it could burst, what does tell you?
With fixed the combined law reduces to , so pressure climbs linearly with kelvin temperature without bound — the ideal model never predicts a burst; the material fails.
What happens to the combined law if two gases at different states are mixed into one container?
It does not apply directly — changed by combining samples, so you must use with the summed moles, or partial pressures per gas.
Is there any temperature (in Celsius) where the Celsius and Kelvin mistake accidentally gives the right ratio?
No — the offset of shifts every ratio , so unless (never), Celsius ratios are always wrong.
If pressure is expressed as a gauge reading, can it go straight into ?
No — the equation needs absolute pressure; gauge pressure omits atmospheric pressure, so you must add first.
Connections
- Boyle's Law — the constant- trap ("directly proportional" error)
- Charles's Law — the Celsius-vs-kelvin trap
- Avogadro's Law — the equal-mass-vs-equal-moles trap
- Kinetic Theory of Gases — WHY pressure responds to squeezing and heating
- Real Gases and van der Waals Equation — WHERE the ideal model breaks
- Dalton's Law of Partial Pressures — mixtures and partial-pressure edge cases