A stainless steel chamber contains Ar gas at a temperature T and pressure P. The total number of Ar atoms in the chamber is n. Now Ar gas in the chamber is replaced by CO₂ gas and the total number of CO₂ molecules in the chamber is n/2 at the same temperature T. The pressure in the chamber now is P′. Which one of the following relations holds true? (Both the gases behave as ideal gases)
- (a)P′ = P
- (b)P′ = 2P
- (c)P′ = P/2
- (d)P′ = P/4
Correct — C, P′ = P/2. Write the ideal gas law in the form that counts particles rather than moles: PV = NkT, where N is the number of atoms or molecules and k is Boltzmann's constant. The chamber is a rigid stainless-steel vessel, so V does not change, and the question holds the temperature at T throughout. With V and T both fixed, pressure is proportional to N and to nothing else. Replacing n argon atoms by n/2 carbon dioxide molecules halves the particle count, so it halves the pressure: P′ = P/2. The identity of the gas never enters. That the CO₂ molecule is heavier than the argon atom, and triatomic rather than monatomic, is exactly the irrelevance the question is planted to test — Avogadro's principle says equal numbers of particles at the same volume and temperature exert the same pressure whatever they are made of.
- (a)P′ = P — This would require the number of particles to be unchanged. It has been halved. A student reaches this answer by reasoning that the heavier CO₂ molecules hit the walls harder and make up for there being fewer of them — but at the same temperature the two effects cancel exactly, which is what makes pressure independent of molar mass.
- (b)P′ = 2P — The wrong direction altogether. Fewer particles in the same volume at the same temperature means fewer collisions with the walls each second, so the pressure must fall, not rise.
- (d)P′ = P/4 — Halving applied twice — once correctly for the particle count and once again for some supposed effect of the change of gas. The molar mass of the gas does not appear anywhere in PV = NkT, so there is no second factor to apply.
The ideal gas law can be written PV = nRT, counting in moles, or PV = NkT, counting individual particles, with k = R ÷ N_A linking the two. Either form says the same thing: the pressure a gas exerts depends on how many particles are present, how much room they have and how hot they are, and on nothing else. The kinetic-theory picture behind it is that pressure is the average effect of particles striking the walls, and at a given temperature every gas has the same average translational kinetic energy per particle.
The examiner's technique here is to load the stem with information that does not matter — the vessel is stainless steel, the gas changes from a monatomic noble gas to a triatomic molecular one — and then hide the single relevant change in the middle. Strip the problem down before calculating: what is held constant, what actually changes? Here V is constant because the chamber is rigid, T is constant because the question says so, and N goes from n to n/2. With two of the three variables pinned, the ratio follows in one line. Heavier molecules do move more slowly at a given temperature, but they carry more momentum per collision, and the two effects cancel precisely — which is the physical content of Avogadro's principle.
- The ideal gas law is PV = nRT in moles, or PV = NkT in particles; at fixed V and T, pressure is proportional to the number of particles.
- By Avogadro's principle, equal volumes of any ideal gases at the same temperature and pressure contain equal numbers of particles.
- Pressure does not depend on the molar mass of the gas — heavier particles move more slowly but strike harder, and the effects cancel.
- Boltzmann's constant k equals R ÷ N_A, about 1·38 × 10⁻²³ J K⁻¹.
- A rigid container means an isochoric change, in which volume is held constant and pressure and temperature vary together.
The switch from argon to carbon dioxide is a red herring — option (c).
- Bringing the molar mass of the gas into a pressure calculation, where it has no place.
- Missing that a rigid metal chamber means the volume is fixed.
- Halving twice and arriving at P/4.
NDA sets the ideal gas law as a ratio problem — one or two quantities changed, the rest held fixed — with an irrelevant change of gas or of container material planted in the stem.
If we plot a graph between volume V and inverse of pressure P (i.e., 1/P) for an ideal gas at constant temperature T, the curve so obtained is
- (a) straight line
- (b) circle
- (c) parabola
- (d) hyperbola
Answer(a) straight line
The same equation approached graphically, and a reminder that these items are all solved by fixing which variables the problem holds constant.
- practice — not a real PYQ
A rigid vessel contains an ideal gas at pressure P and temperature T. If the absolute temperature is doubled while the number of molecules and the volume stay the same, the new pressure is
- (a)P/2
- (b)P
- (c)2P
- (d)4P
Answer(c) 2P — at fixed volume and particle number, PV = NkT makes pressure proportional to absolute temperature.
- practice — not a real PYQ
Two identical rigid flasks at the same temperature contain equal numbers of molecules, one of hydrogen and one of oxygen. The pressures in the two flasks are
- (a)equal
- (b)greater in the hydrogen flask
- (c)greater in the oxygen flask
- (d)in the ratio 1 : 16
Answer(a) equal — by Avogadro's principle, pressure depends on the number of particles and not on their mass.