# Thermodynamics - Temperature

1. Jan 5, 2005

### Kelvin

Hi, I am taking a thermodynamics course but I am confused when my teacher told me that "the temperature of a non-equilibrium system is undefined".

but as I know, in microscopic world, temperature measures the average kinetic energy of a system, and this seems to be inconsistent with the previous statement.

Can anybody point out the mistakes I have made? Thank you!

2. Jan 5, 2005

### Q_Goest

Would your teacher consider a "non-equilibrium system" to be a red hot chunk of metal tossed into a bucket of cold water? The system then consists of the hot metal and the cold water. We might define the temperature of that system as the equilibrium temperature, but that temperature doesn't exist in the system anywhere.

3. Jan 5, 2005

### Palindrom

2 different things

All thermodynamic parameters are defined only in Equilibrium State. You can talk of temperature in quazisatatical processes, but not much more than that.
The sentence "Temperature measures the average kinetic energy of a system", is helpull when you try to understand what you're dealing with, but you can't use it in equations. Temperature is really defined as the Internal Energy's partial derivative by the Entropy.

4. Jan 5, 2005

### Kelvin

Is "the partial derivative of internal energy by entropy" is a more fundamental definition of temperature? Is the expression
KE = 3/2 kT​
meaningful when the system is in equilibrium? thermodynaimcs parameters are pressure, volume & temperature only?

Q_Goest:
Is that means my teacher was wrong or probably I made some mistakes when jotting notes?

5. Jan 5, 2005

### Andrew Mason

KE=3/2kT is an average KE for one molecule in a system at temperature T. That doesn't mean the molecule has that temperature. Temperature is a statistical measure and only applies to large numbers of molecules. In any gas, a given molecule can be moving at a wide range of speeds because it is subject to random collisions.

But I disagree that temperature is not defined for a non-equilibrium system. The fact that heat is flowing into or out of a system does not mean it has no temperature. A cooking turkey, for example is not in thermal equilibrium. It still has a defined temperature and if you stick a thermometer probe into it while it is cooking, you can measure it.

AM

6. Jan 5, 2005

### dextercioby

Yes,u can bring into discussion temperature only in equilibrium states,or in states not far from the equlilibrium states.That T in the eq.
$$T=\frac{2}{3}\frac{E}{k}$$
is the same temperature with the one dicussed in the thermodynamics of equilibrium processes.It is called KINETIC TEMPERATURE AND IS DENOTED BY
$$\Theta =:\frac{2}{i}\frac{E}{2}$$
.However,because this kinetic temperature,in the case of statistical systems in equilibrium,coincides with the absolute termodynamic temperature (denoted by T and measured in Kelvin),it is denoted like the latter,viz.with T.
Temperature is a statistical quantity.That's because entire thermodynamics of reversible/equilibrium processes (in either formulation,but the neogibbsian is more easy to use) contains the same results as a subtheory of statistical physics of equlibrium processe called 'statistical thermodynamics'.
In the neogibbsian formulation of thermodynamics,temperature is defined implicitely by:
$$\frac{1}{T}(U,\{X_{i}\})=:(\frac{\partial S(U,\{X_{i}\})}{\partial U})_{\{X_{i}\}$$ (1)
,which is bsically the same with its definition within the microcanonical ensemble (classical or quantum) of statistical mecanics of equilibrium processes:
$$\frac{1}{T}(E,\{X_{i}\})=:(\frac{\partial S(E,\{X_{i}\})}{\partial U})_{\{X_{i}\}$$ (2)
,where E is the value of the Hamiltonian,assumed fixed at macroscopical level.It is actually the internal energy from thermodynamics.

Daniel.

Last edited: Jan 5, 2005
7. Jan 5, 2005

### krab

As was stated, temperature is a statistical quantity. If you try to attach T to individual particles (actually, degrees of freedom not particles), you'll get widely differing results and no useful quantity. But large numbers of particles do have a temperature. The number of particles needed for a useful quantity is related to how much "noise" you can tolerate. If you want a temperature correct to 1%, you only need about 10,000 particles. (Noise ~ 1 / square root of N) This is quite a small number on macroscopic scales of 10^24 particles. So though your teacher may be correct in some sense, it requires a measurable temperature gradient across a volume containing only some 10,000 particles before we can say the medium is too far from equilibrium to define a temperature.