To determine which of these compounds is named without a Roman numeral, let's look at each one individually and understand the naming conventions for each specific case.
1. \( \text{Cr}_2\text{O}_3 \) (Chromium(III) oxide):
- Chromium is a transition metal and can have multiple oxidation states. For this compound, the oxidation state of chromium is +3, so its name includes the Roman numeral (III) to indicate the specific oxidation state. Therefore, this compound is named Chromium(III) oxide and includes a Roman numeral.
2. \( \text{Fe}_2(\text{CO}_3)_3 \) (Iron(III) carbonate):
- Iron is also a transition metal and can have multiple oxidation states. In this compound, iron is in the +3 oxidation state. Its common name is Iron(III) carbonate with the Roman numeral (III) indicating the oxidation state. Therefore, it includes a Roman numeral.
3. \( \text{Hg}_2\text{S} \) (Mercury(I) sulfide):
- Mercury can exist in +1 and +2 oxidation states. In this compound, the oxidation state of mercury is +1. The name of the compound is Mercury(I) sulfide, which includes the Roman numeral (I) to indicate the oxidation state. Therefore, it has a Roman numeral in its name.
4. \( \text{K}_2\text{SO}_4 \) (Potassium sulfate):
- Potassium is an alkali metal located in Group 1 of the periodic table. Alkali metals typically form only one type of ion, which is \( \text{K}^+ \). Therefore, there is no need to specify the oxidation state with a Roman numeral in its name. The compound is simply named Potassium sulfate, which doesn't use a Roman numeral.
Thus, the compound named without a Roman numeral is:
[tex]\[ \boxed{K_2SO_4} \][/tex]
