The entropy, you may have negative entropy values as well as positive value according to following,
S=K ln w( Omega)
Where K=boltzmann’s constant and omega is the number of microstates corresponding to given state the answer is No,
Why omega the number of microstate possible for a certain state can never be smaller than one, Since Boltzmann’s constant is a positive number and Ln omega will always be greater or equal to zero , Entropy will never be negative.
However when calculating ∆S (change in entropy is a thermodynamic Process), Yes Entropy can be negative. Remember entropy is essentially the state of disorder of a system(on Macroscopic Level) The natural progression of the world is from order to disorder.(e.g. there are more ways to have a messy room than to have an impeccable neat room). For the change in entropy to be negative. Just think of it in terms of the room analogy, initially, it was messy but then it goat neater. The state of disorder of things was lessened. Applying this to a chemistry example
CO2(gas) = CO2(S) an element(compound ) is gaseous state always has greater state of entropy(gaseous molecules are more free to move. However , an element/ compound in a solid state has a smaller state of entropy. Because molecules in solid state are less free to move. Smaller state of entropy greater state of entropy =negative entropy.
"A negative value of ΔS* suggests that the adsorption process involves an
associative mechanism. The adsorption leads to order through the formation of an activated complex between the adsorbate and adsorbent. Also a negative value of ΔS* reflects that no significant change occurs in the internal structures of the adsorbent during the adsorption process".