# The difference between heat and temperature

So as I understand it, heat energy of an object is the SUM of all the kinetic energies of the molecules of the object (upto constant factor).

The temperature on the other hand is the AVERAGE of the kinetic energies of all the molecules of the object.

Now when ice is melting at 0 degrees Celsius, the temperature as measured on a thermometer does not go up.

The common explanation is that any heat being absorbed by the ice is being used to break the somewhat strong solid solid bonds between the molecules of the ice.

Here is my question. if heat of an object is what I defined above, then since all the molecules are increasing in kinetic energy, the average of the kinetic energy should also increase, meaning the temperature should increase. But at 0 degrees celsius for ice that does not seem to be the case.

Where am I going wrong in my understanding?

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" A bathtub filled with lukewarm water contains more heat than a red-hot nail." –  Mark Rovetta Jun 4 '13 at 1:21

Heat is not a property of a system. Heat is a process function. Temperature is a property of a system because is a state function. For instance, the state of a simple gas is given by temperature, pressure, and composition $(T,p,N)$.

Temperature is defined as the inverse ratio of variation of entropy $S$ to changes in internal energy $U$ $$T \equiv \left( \frac{\partial S}{\partial U} \right)^{-1}$$ This is the thermodynamic concept of temperature, which is more general than the kinetic concept that you are considering. Regarding your question, part of the energy given as heat is used to break the bonds and when are broken if you continue supplying energy this will increase the kinetic energy of the molecules.

Moreover, kinetic temperature is not the average of the kinetic energies of all the molecules of the object. This average of kinetic energies is the average kinetic energy. The kinetic temperature is defined as $2/3$ the average internal energy per number density.

At the other hand, heat $Q$ is defined for a given process as the internal energy interchanged which is neither work nor due to flow of matter $$Q \equiv \Delta U - W - U_{matter}$$ Notice that internal energy is a state function and $\Delta U$ denotes the difference between the initial and final energies, but heat is not a state function and this is why we write $Q$ instead of an incorrect $\Delta Q$.

The concept of process function is most easily understood with the example of a lake. A lake has some amount of water, and this can change by evaporation and raining. You can count the amount of water added to the lake by some raining process, but the lake itself does not have any amount of "raining" or evaporation" only some amount of water. Similarly a thermodynamic system has internal energy but has not heat or work.

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The kinetic energy of the atoms is only the nuclear part of the internal energy. The electronic structure responsible for chemical bonds is another part.

Any heat used to change the electronic structure (i.e., breaking bonds) does not affect the kinetic energy of the atoms.

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Why does temperature of ice not increase when heat is supplied to ice?

Breaking of hydrogen bonds when ice melts is an endothermic reaction. Energy in the form of heat is absorbed to break the hydrogen bonds in this reaction. Kinetic energy of molecules does not increase. But the internal energy of the ice/water increases.

source: http://en.wikipedia.org/wiki/Ice (see characteristics)

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A different perspective might help to clear the confusion:

The average does not have to change. During melting you just "add" molecules to the liquid part of your system. While the number of water molecules increases, the temperature during this process and the kinetic energy of the individual molecules stays constant. Only if all ice is melted away all molecules are in the same thermodynamic phase. The energy is used up to break the bonds.

Now heating up the system further increases the average of the kinetic energy. So there is not really a paradox here and your descriptive definition is not wrong either. You only neglected that this is a first order phase transition with two different phases of matter co-existing.

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heat is the total kinetic energy possessed by all randomly vibrating particles of a body. heat is an energy as well as process function.It cause the temperature to increase.

Temperature is the average rate of vibration of particles due to heat.It is a unit as well as state function.It is measured by the help of themometer.

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You mix things up, please check your textbook. –  hwlau Sep 4 '13 at 0:44
This is a reasonable description of the internal energy of certain (intentionally simple) system. But internal energy is not heat. Heat, like work, is a measure of energy transfer. –  dmckee Sep 4 '13 at 1:55