I don't understand q = mc delta T

[nhmllr]

Well, I understand q = mc∆T, along with q = mH_v and q = mH_f

What I don't understand is this graph:
http://dinosaurtheory.com/phase_change.jpg

Well, I mean, I understand THAT graph.

Here's what I don't understand:
Today in chemistry, we received a very similar graph, but the X-axis was labeled "time" instead of "added heat" as it is in the one I linked to.

I raised my hand and pointed out that the independent variable should not be "time" because nowhere in our equations were we even given "time" as a variable! "Time" actually has nothing to do with temperature. I could leave an ice cube on my table and it won't necessarily melt, then vaporize. Or, alternatively, I could throw it into the sun and it would vaporize very quickly. Also, over time I could have water vapor condense into water, then freeze into ice.

The way I see it, "time" has no right to be and independent variable on such a graph.
My chemistry teacher told me that "time" is often an independent variable in graphs, and that energy had to be calculated (meaning that added energy could not be the independent variable if we had to calculate it after the fact). But I could not get over the fact that nowhere in any equation was "time" mentioned, and I couldn't get over that.

My Question: Is time supposed to be the independent variable on this graph? Or is my thought process right in that the X-axis should be labeled "added energy?"

(I understand that time can be the X-axis with the given condition that we're heating something up over time, such that heat = constant x time, but nowhere were we given such a condition.)

 

[jtbell]

If heat is added to the system at a constant rate (calories per second or joules per second), then a graph of temperature versus time looks the same as a graph of temperature versus added heat, except for the horizontal scale.

 

[boneh3ad]

It wouldn't even have to be a constant rate for time to be a valid variable on that axis. It would still need to be constant rate to qualitatively match the graph, of course.

 

[Philip Wood]

heat = constant x time

This equation says it all. Plotting time instead of heat added will give you the same shape of graph. Don't worry about your teacher not mentioning it. You clearly understand what's going on.