# How many Degrees of Freedom do Linear Triatomic Gas Molecules Have and Why?

I read this in one of my textbooks that stated

Linear Triatomic Gas Molecules like CO2, HCN, CS2 in which all the atoms are linear, the total number of degrees of freedom are

Nf = NT + NR + NV where NT, NR, NV are the translational freedom, rotational freedom and vibrational freedom/s respectively.

Is this correct ? If yes then why do Linear Triatomic Gas Molecules have 7 degrees of freedom, If no then what will be the correct degree/s of freedom and why ?

If your textbook is on thermal physics, then a linear triatomic molecule is said to have $$3$$(translational)$$+2$$(rotational)$$+3*3-5$$(vibrational)=$$9$$ d.o.f. For all vibrational modes to be active requires high temperatures. So normally at moderate temperatures, only $$2$$ vibrational modes are active. That's why people talk of $$7$$ d.o.f for a linear triatomic molecule.
For a linear triatomic molecule, there are 3 degrees of freedom for each molecule, i.e., a total of $$3N = 3 \times 3 = 9$$ degrees of freedom. Since there are 2 constraints (say, bonds) between the 3 molecules, the total degrees of freedom are now $$3N - 2 = 7$$ degrees of freedom. Out of which 3 are translational, 2 are rotational, and the rest 2 is vibrational.