Figure 12.8 shows plot of \(\frac{PV}{T}\) versus P for 1.00×10–3 kg of oxygen gas at two different temperatures.
(a) What does the dotted plot signify?
(b) Which is true: T1>T2 or T1<T2 ?
(c) What is the value of \(\frac{PV}{T}\) where the curves meet on the y-axis?
(d) If we obtained similar plots for \(1.00×10^–3\) kg of hydrogen, would we get the same value of \(\frac{PV}{T}\) at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of \(\frac{PV}{T}\) (for low pressure high temperature region of the plot) ? (Molecular mass of H2 = 2.02 u, of O2 = 32.0 u, R = 8.31 J mo1–1 K–1.)
A plot of PV/T versus P for 1.00 × 10−3 kg of O2 at two temperatures T1 and T2.
(a) Significance of the dotted line
The dotted horizontal line represents the ideal gas behaviour.
For an ideal gas:
PV = nRT ⇒ PV/T = nR = constant
Thus, the dotted line corresponds to the value of nR, which is independent of pressure.
(b) Which is true: T1 > T2 or T1 < T2?
At higher temperature, real gases behave more ideally and deviations from the dotted line are smaller.
From the graph, the curve labeled T2 shows smaller deviation from ideal behaviour than T1.
Therefore:
T2 > T1
(c) Value of PV/T where the curves meet the y-axis
At the y-axis, pressure P → 0, and the gas behaves ideally.
Hence:
PV/T = nR
Mass of O2 = 1.00 × 10−3 kg = 1.0 g
Molar mass of O2 = 32 g mol−1
Number of moles:
n = 1.0 / 32 = 0.03125 mol
Using R = 8.31 J mol−1 K−1:
PV/T = nR = 0.03125 × 8.31
PV/T = 0.26 J K−1
(d) Comparison with hydrogen gas
For hydrogen, PV/T at low pressure is also equal to nR.
For the same mass (1.00 × 10−3 kg), hydrogen will have a different number of moles due to its lower molar mass.
Molar mass of H2 = 2.02 g mol−1
Moles of H2 in 1.0 g:
n = 1.0 / 2.02 = 0.495 mol
Thus, PV/T = nR would be much larger, so the value would not be the same.
Mass of H2 required to give same PV/T
For same PV/T:
nH2 = nO2 = 0.03125 mol
Mass of H2:
= 0.03125 × 2.02
= 0.063 g = 6.3 × 10−5 kg
Final Answers:
(a) Dotted line represents ideal gas behaviour (PV/T = nR).
(b) T2 > T1.
(c) PV/T = 0.26 J K−1.
(d) No, hydrogen will not give the same value for the same mass.
Mass of H2 required = 6.3 × 10−5 kg.
