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Is the only thing that remains the same at equilibrium between two different size samples of the same reactants and products, the equilibrium expression (kc)? How come the ratio without the exponents changes but the ratio with the exponents doesn't change?
The important distinction is that the ratio with the exponents is the equilibrium constant, or some variation of it (one of them is also the reciprocal of K). This means that these specific ratios are unchanged across different samples sizes given the temperature is constant. Because the ratio without the exponents is not the equilibrium constant and is only the ratio between products and reactants, it doesn't stay the same. Another way to think about this is to plug in values; they don't really have to be accurate, they're only for the sake of showing different ratios of products/reactants depending on the initial sample sizes (assuming you do this, the ratios do end up being different for the two separate samples).
The equilibrium constant stays the same at equilibrium because it's a constant: it's a calculated factual value for that exact reaction at the given temperature. The ratio with the exponents is the equilibrium constant, so it will stay the same. And like Vincent said, the other ratio you mentioned is subject to change with a change in amount of the reactants or products.
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