# How are Entropy and Temperature conjugate variables?

1. Aug 17, 2013

### 541099

Can someone conceptually explain to me how Temperature and Entropy are conjugate variables?

I would imagine that Temperature and Internal Energy would be more appropriate, as I understand Heat flow causes changes in Internal Energy, some of which is used to change the translational motion of a particle, while the rest is used to increase the Entropy of the particle.
The portion that is increases translational motion is effective used to do Work, while the portion that increases Entropy represents a loss of Thermal Energy.

Do translational motion and entropy not constitute variables that fall under Internal Energy?

2. Aug 19, 2013

### Khashishi

Temperature is defined as $\frac{1}{T} = \frac{\partial S}{\partial U}$
This gives $T = \frac{\partial U}{\partial S}$

Do you understand why pressure and volume are conjugate variables? Temperature and entropy appear analogous when the equation of state is written.
$dU = T dS + P dV$

The pressure tells you how hard (how much energy it takes) it is to change the volume. The temperature tells you how hard it is to change the entropy.

3. Aug 20, 2013

### 541099

Khashishi, I guess what's confusing me is that Temperature is defined as the average kinetic energy of a particle in a system. This seems to necessitate the ability to do work.
If an increase in Temperature elicits an increase in Internal Energy, some would be manifest as an increase in Pressure by virtue of the inherent increase in translational motion (which would increase PV-Work), while some would inevitably be used to increase the Entropy (which is simply the non-translational components of molecular motion, from what I understand). Please correct me if I am misunderstanding anything.

I suppose my confusion is also compounded by my ignorance as to the relationship of Temperature to Energy transfer via Heat and Work. It seems to me that Temperature quite literally is a measure of the amount of Work a particle can do through its Kinetic Energy. Ironically, I've lost perspective of how Temperature is associated with Thermal Energy.

4. Aug 20, 2013

### DrDu

You are duely confused. Noone defines temperature this way.
Temperature defines the direction of the flow of heat. Heat flows always from the system with higher temperature to the one with lower temperature. For such a definition of temperature being possible you need the zeroth law of thermodynamics which states transitivity of heat flow.

5. Aug 21, 2013

### Khashishi

If you want to do thermodynamics, you have to use the proper thermodynamics definition of temperature. I've already given you the definition above.

You might wonder, why is it defined this way? Well, consider two bodies that are at different temperatures that are placed in contact. In equilibrium, the temperatures are equal. Why does heat flow from the hotter body to the cooler body? An interpretation of the second law of thermodynamics is that the entropy increases as the system approaches equilibrium. If body A gains more entropy per unit heat (energy) than body B, then we call body B "hotter" than body A, because heat will flow from B to A to maximize the entropy. Therefore, temperature is a less fundamental concept than energy or entropy, and can be understood in those terms.

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