What happens to the temperature of a material when it is undergoing a change of state?

Typically nothing, but read further.

Typical changes of state (i.e. melting, freezing, condensing, boiling, sublimation, deposition) are equilibria of either constant-pressure, constant-temperature, or neither.

Example phase diagram:

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(Note that the image has an error. `"solid" -> "gas"` is called sublimation, not sublimination.)

The change in temperature is `0` throughout a vertical phase transition (such as gas to liquid), but the change in pressure is `0` for a horizontal phase transition.

You can also perform a diagonal phase transition if you vary the pressure AND temperature.

Finally, if you land on exactly the boiling/melting point, and you keep the temperature AND pressure constant, a natural phase transition occurs, such as everyday boiling or melting.

In either case, you may find it useful that what we have is the Maxwell Relation

`\mathbf(DeltaG = -SDeltaT + VDeltaP).`

This means for constant-temperature (vertical) phase transitions (e.g. isothermal compression/expansion) in a closed system, we have for the Gibbs' free energy

`DeltaG_"trs" = V_"sys"DeltaP_"sys"`

Or for constant-pressure (horizontal) phase transitions, we have

`DeltaG_"trs" = -S_"sys"DeltaT_"sys"`

Or, for constant-temperature, constant-pressure phase transitions (such as everyday boiling/melting), we have

`DeltaG_"trs" = 0, => DeltaH_"trs" = T_"sys"DeltaS_"trs"`

Or, for a phase transition in which neither is constant (diagonal transition):

`DeltaG_"trs" = -S_"sys"DeltaT_"trs" + V_"sys"DeltaP_"trs"`