How to distinguish between empirical formulas and first-principles models?

Haha, that's an excellent question, and one that many people easily confuse. I'll do my best to explain my understanding in plain language.

You can think of it this way:

Empirical Formula = The 'Trade Secret' of an Experienced Pro

An experienced old carpenter doesn't need to calculate complex mechanics; just by feel and sight, he knows how much force to apply and from what angle to split a piece of wood most efficiently and effectively.

If you ask him why, he might not be able to articulate a complete physical theory, but he'll tell you: "I've been doing this for decades, and this is just how it's done!"

This "trade secret" is the empirical formula. It's derived from extensive observation, experimentation, and accumulated experience.

• Characteristics: It tells you "what" and "how," but rarely "why."
• Advantages: Within the scope of its experience, it's usually very accurate, simple to use, and fast.
• Disadvantages: If the situation changes and falls outside its experienced range (e.g., a type of wood he's never seen before), this "secret" might fail. It lacks strong generalization ability. What we often call "fitting parameters" or "cobbling together formulas" often falls into this category.

First-Principles Model = A Physicist's 'Scientific Derivation'

A physicist also wants to solve the problem of splitting wood. He wouldn't start by splitting wood ten thousand times to get a feel for it.

He would start from the most fundamental principles:

1. Study the wood's internal fiber structure, density, and moisture content (these are fundamental properties of matter).
2. Analyze the axe's weight, material, and blade sharpness.
3. Apply Newton's laws of motion (F=ma), material mechanics, and other most basic physical laws.

Using these foundational, universally accepted scientific laws, he builds a mathematical model to calculate how to split wood most efficiently under specific conditions.

This process of deriving from the "source" is the first-principles model.

• Characteristics: It attempts to answer the most fundamental "why." The entire model contains almost no arbitrarily chosen parameters; everything is derived from basic laws.
• Advantages: Because it's based on universal scientific laws, its predictive power is very strong. Even for wood never seen before, as long as you know its basic physical parameters, this model can tell you how to split it. It has good generality/universality.
• Disadvantages: The process is extremely complex, with massive computational demands. Sometimes, approximations are made for simplification, but the core idea remains unchanged.

In Summary:

Simply put, an empirical formula is like "copying homework," summarizing the patterns of all past correct answers; whereas first principles is like "solving the problem," deriving the answer step-by-step using the most basic formulas.