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I'm not Chemist. I'm taking a chemistry class in college, and the test review contains many questions where I'm asked to rank from least to greatest based on bond order, polarity, and bond angles. I feel like I can rate on electronegativity.

The problem is that I don't know a way to rationalize this in my mind without drawing the Lewis Structures. Many times each possible choice has like 5 or so, so it takes a lot of time. We are only given 50 minutes for the test and there are many more problems that require large amounts of time.

As a chemist or someone who is more knowledgeable in this field than I am, are there any techniques that would help you now or when you were first learning?

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Although it is often taught otherwise, hybridisation is a result from molecular structure. Therefore you must know (or be able to predict) the structure. On a basic level, the easiest way to do that are Lewis structures.
A different way is using the VSEPR theory, a rudimentary model to predict shapes and molecular geometries. Be warned though that this model will fail when it comes to more complex molecules. You need a good imagination of the three dimensions to be able to do it without drawing it.
However you want to spin it, you will have to know which atoms are bonded to other atoms. If you can imagine that, that is good, if not you will have to draw it out. The longer you study, the more present some common molecules are and you will be able to tell certain properties by what we like to call chemical intuition. But that takes time.

Bond order is a bit of a tricky topic, because you have to know the bonding situation. This would include all possible resonance structures for some molecules. While knowing or judging this also comes with experience, it is probably best to draw such structures for learning purposes.

I personally do not like exam questions like these, because they usually just test if a student is good at cramming facts, rather than asking about if these things are understood. But things will get better, if you don't lose interest in the mean time.

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