# Temperature defined in terms of entropy and energy

1. Feb 4, 2012

### jd12345

According to wikipedia :- " Formally, temperature is defined as the derivative of the internal energy with respect to the entropy."

I always thought entropy is defined in terms of energy and temperature.
IS this true that temperature is defined in terms of entropy and energy?

According to it :- dS = dQ / T
IS this somehow related to the fact that lowest temperature can only be zero as then entropy change will be zero
I found all this connected but somehow cannot put them together. Is kelvin scale defined with the help of this equation???

Last edited: Feb 4, 2012
2. Feb 5, 2012

### juanrga

i)
The more general definition of temperature is
$$T \equiv \left( \frac{\partial S}{\partial U} \right)^{-1}$$
The definition given by the Wikipedia is not so general, although is equivalent for equilibrium.

ii)
Entropy is not defined in terms of energy and temperature but $S=S(U,V,N...)$ Temperature T is not one of the natural variables of entropy (Gibbs–Duhem equation).

iii)
The expression $dS = dQ / T$ is only valid for closed systems under reversible process. For irreversible process and/or open system this expression needs to be generalized. Different extensions of the above equation for open systems are available. See Non-Redundant and Natural Variables Definition of Heat Valid for Open Systems for a survey of the definitions used by Callen, Casas-Vazquez, DeGroot, Fox, Haase, Jou, Kondepudi, Lebon, Mazur, Misner, Prigogine, Smith, Thorne, and Wheeler.

For instance using Kondepudi and Prigogine definition the expression is (see eq. 12)
$$dS = d_iS + d_eS = d_i S + dQ / T + s_k dn_k$$

iv)
Entropy is a non-decreasing function of energy. Therefore the partial $\partial S/ \partial U$ is always non-negative and, thus, temperature is positive or zero.

v)
The kelvin, is the SI unit of thermodynamic temperature, and is defined as «the fraction 1/273.16 of the thermodynamic temperature of the triple point of water».

Last edited: Feb 5, 2012
3. Feb 6, 2012

### Amir Livne

@juanrga

I can vaguely recall from my Thermo course some time ago that there are discrete systems, where you have N particles that can be in either base or excited states. In those systems once the energy is >N/2, temperature is negative.
Am I right here? Or is it an axion that the derivative is positive?

4. Feb 7, 2012

### juanrga

That the derivative is non-negative is one of the postulates in Tisza-Callen axiomatic formulation of thermodynamics.

The so-named negative temperatures arise from a formal analogy, with the definition given above for thermodynamic temperature, when you consider a kind of 'exotic' systems for which the derivative is negative. But those systems are not equilibrium systems and those formal negative temperatures are only valid in the first nanosecond before the excited particles spontaneously decay.

Moreover, those formal negative temperatures have to be interpreted as hotter than infinite positive temperature, which is not easy to understand.

Last edited: Feb 7, 2012
5. Feb 7, 2012

### Ken G

The place where you see that kind of thing in a steady state is in a lasing cavity, where some form of ongoing pumping induces the negative-temperature population inversion. Of course that is not a strictly thermodynamic temperature because it refers only to the entropy of the subsystem doing the lasing and not whatever is doing the pumping, so that may be how it escapes the more formal axiomatization.