When can the molecule 2-butene, #C_4H_8#, undergo a geometric change called cis-trans isomerization?

The cis-trans isomerization of but-2-ene requires breaking of the π bond.

The minimum energy required for isomerization is 267 kJ/mol.

Thermal Isomerization

At temperatures of about 500 °C the molecules have enough kinetic, vibrational, and rotational energy that the π bond breaks and isomerization can take place.

Photochemical Isomerization

A quantum of light can excite a π electron to a π* orbital and allow free rotation about the central C-C bond.

(from www.tutorvista.com)

In the excited π* state, the double bond is effectively reduced to a single bond, and one methyl group can rotate relative to the other.

The isomerization can be accomplished by providing another pathway with a lower activation energy (catalysis).

Acid-Catalyzed Isomerization

In the presence of strong acids, which act as catalysts, the isomerization can take place at room temperature.

#underbrace("CH"_3"CH=CHCH"_3)_color(red)("cis") stackrel("H"^+)(⇌) "CH"_3stackrel(color(red)(+))("C")"H-CH"_2"CH"_3 stackrel("H"^+)(⇌) underbrace("CH"_3"CH=CHCH"_3)_color(red)("trans")#

The proton adds reversibly to the π bond forming a singly-bonded carbocation.

There is a temporary free rotation about the single bond before the proton leaves to regenerate the π bond.

Free-Radical Catalyzed Isomerization

If we add iodine, the temperature required for the isomerization drops drastically.

Iodine sets up an equilibrium with iodine atoms.

#"I"_2 ⇌ "2I·"#

Then the mechanism proceeds as below.

(from faculty.chem.queensu.ca)