First slide

Kinetic theory of ideal gases

Question

We have 0.5 g of hydrogen gas in a cubic chamber of side 3 cm kept at NTP. The gas in the chamber is compressed keeping the temperature constant until a final pressure of 100 atm is reached. What is the volume of gas in final state as per ideal gas assumption? Is one justified in assuming the ideal gas law, in the final state? (Hydrogen molecules can be considered as spheres of radius 1 $\overset{o}{A})$ .

Very difficult

A

$2.7 \times 10^{- 7} m^{3}, N o$
B

$4.7 \times 10^{- 6} m^{3}, Y e s$
C

$4.7 \times 10^{- 5} m^{3}, N o$
D

$4.7 \times 10^{- 4} m^{3}, Y e s$

Solution

The ideal gas law is applicable if the volume of molecules is negligible in comparision to the volume of the gas.
Volume of hydrogen molecule = $V = \frac{4}{3} π {(1 \times 10^{- 10})}^{3} = 4 \times 10^{- 30} m^{3}$
Number of moles of hydrogen = $n = \frac{m a s s o f g a s}{m o l a r m a s s o f g a s} = \frac{0.5}{2} = 0.25$
Number of H₂ molecules present = $N = 0.25 \times (6.023 \times 10^{23})$
Volume of these molecules = $N V = 0.25 \times (6.023 \times 10^{23}) \times (4 \times 10^{- 30}) = 6 \times 10^{- 7} m^{3}$
Using ideal gas equation,
$P_{1} V_{1} = P_{2} V_{2}$
$V_{2} = \frac{P_{1} V_{1}}{P_{2}} = \frac{1}{100} {(3 \times 10^{- 2})}^{3} = 2.7 \times 10^{- 7} m^{3}$ .
The two volumes are close. Hence, intermolecular forces will play role and gas behaviour will deviate from ideal gas.

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