Is Internal Energy Always Proportional to Temperature?

[jack1234]

Internal energy vs temperature(q37)

For this question:
http://tinyurl.com/3ymlnj

The answer to this question is c.

But according to my understanding, increasing of internal energy means increasing of temperature. Hence is isothermal situation, the change of internal energy is 0.
With this understanding, if c is the answer, b will be the answer.

But now b is not the answer, hence I wish to know is there anything wrong with this understanding?

 

[Pythagorean]

conversion of state (for example: from a solid to a liquid) costs energy, and while the energy is going into converting the state of the matter, it doesn't go into raising the temperature of it.

So when we find out how much energy it takes to make -5 degree ice into 7 degrees water, we have to add together:

1) the energy it takes to raise it from -5 to 0 degrees C
2) the energy it takes to convert it from ice to water
3) the energy it takes to raise it from 0 to 7 degrees C

so when matter is at its melting or boiling points (it's temperature of phase change) it can exist in both phases with that same temperature. Ice and water can both exist at 0 degrees C (32 F)