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According to the actual data, van der Waals constant $b$ of $\ce{NH3}$ gas is $\pu{0.0371 L mol-1}$ and that of of $\ce{N2}$ gas is $\pu{0.0387 L mol-1}$. But it should be, $$b_{\ce{NH3}}>b_{\ce{N2}}$$ This is because structure of $\ce{NH3}$ is a tetrahedral while that of $\ce{N2}$ is linear, so excluded volume of $\ce{NH3}$ should be greater when compared to that of $\ce{N2}$. As not much difference in size between them, bond length of $\ce{N-H}$ in $\ce{NH3}$ is $\pu{1.008 Å}$ while that of $\ce{N#N}$ is $\pu{1.097 Å}$, why is the opposite thing happening?

Refer here for data of van der Walls constant b

Refer here for bond length(s)

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    $\begingroup$ For one thing nitrogen atoms are much bigger than those of H. You also have to take into consideration the distribution of electron density, which becomes localized between atoms during bonding. $\endgroup$
    – Buck Thorn
    Commented Jan 3, 2023 at 15:00
  • $\begingroup$ But size should not as much higher because $\ce{N_2}$ has triple bond while $\ce{NH_3}$ has 3 single bond which I think should not as much higher as we think of, and also on the other hand $\ce{NH_3}$ is tetrahedral which should make more volume of sphere of exluded volume. $\endgroup$
    – Leibniz-Z
    Commented Jan 3, 2023 at 15:09
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    $\begingroup$ Have you looked up other information on the relative dimensions of these molecules? For instance bond length? Since that is instrumental to your argument, please do the geometric calculations and include them into the post. $\endgroup$
    – Buck Thorn
    Commented Jan 3, 2023 at 15:29
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    $\begingroup$ Be aware that van der Waals equation is not a tool to experimentally determine sizes of molecules. It is just a mathematical model. Note that there is strong correlation between values of both a and b parameters, with relation to critical p and T. So values of parameter b in context of the molecule effective size are not easily comparable between molecules of very different behaviour in context of intermolecular forces. $\endgroup$
    – Poutnik
    Commented Jan 3, 2023 at 15:56
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    $\begingroup$ You might now account for the atomic radii to determine the longest end-to-end distance in the molecule. For NH add the atomic radius of H at one side, On the other side you need to consider the effect of the angle in the tetrahedral molecule. For NN add twice the atomic radius of N. (this is highly approximate of course but a starting basis for a comparison of predicted geometries). For radii see for instance en.wikipedia.org/wiki/Atomic_radius#Calculated_atomic_radius $\endgroup$
    – Buck Thorn
    Commented Jan 3, 2023 at 16:01

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From a purely geometrical standpoint and using some simple assumptions the similarity and order of b parameters should not be all that surprising.

For ammonia the longest distance in the molecule can be taken as the sum of the length of one NH bond plus the H radius, plus the projection at an angle of $73^ \circ$ of that same distance ($r_\text{H}+r_\text{NH}$) on the opposite side of the N center. This gives $d_\ce{NH3} = 1.9877 Å$.

For dinitrogen we can sum the bond length and twice the computed radius of a nitrogen atom, arriving at $d_\text{NN}= 2.217 Å$.

The ratio $d_\ce{NN}/ d_\ce{NH3} = 1.1154$ compares favorably with the alternative estimate $(b_\ce{NN}/ b_\ce{NH3})^{1/3} = 1.0142$ using the van der Waals parameters as estimates of (or proportional to) the molecular volume. The latter could be argued to be smaller because it represents an effective radius, an average of interactions along different angles of approach rather than only along the longest distance in the molecule.

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    $\begingroup$ So, what if someone didn't knew the data? What will be the factor to decide it? $\endgroup$
    – Leibniz-Z
    Commented Jan 4, 2023 at 7:38
  • $\begingroup$ @Siddharth You chose a difficult example. The b parameters (and molecular volumes) are very similar. For some cases the difference in molecular volume is more obvious and decisive. For others (eg acetonitrile vs benzene!) it's not (the difference in b is negligible but one would expect a large difference in volume). In general regard b as determined by volume but potentially also interactions (consider comment by Poutnik). It is empirically determined (experimental). $\endgroup$
    – Buck Thorn
    Commented Jan 5, 2023 at 12:26

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