Understanding the Role of Fundamental Temperature in Thermodynamics

In summary: I see...so does that make it unreasonable to speak of the equilibrium states of an isolated system?No, but it seems unreasonable to speak of the temperature of a system whose energy could not be altered, even in theory.
  • #1
Gear300
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The fundamental temperature is defined so that 1/τ = ∂σ/∂U. This relation occurs as an equilibrium state, so wouldn't that imply that ∂σ/∂U = 0, leaving the temperature undefined?
 
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  • #2
Gear300 said:
This relation occurs as an equilibrium state, so wouldn't that imply that ∂σ/∂U = 0

Why?
 
  • #3
∂σ/∂U = 1/g(∂g/∂U), in which g is the multiplicity of states. Let us say that the system is uncoupled, then wouldn't the equilibrium condition imply ∂g/∂U = 0 (or is it that temperature is defined specifically for a coupling between systems)?
 
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  • #4
I think I see what you mean. I suppose one would have a problem defining temperature for a system that one absolutely could not add energy too, even in theory. Then ∂σ/∂U would have no meaning, U being constant. But every system dealt with in practice can conceivably be heated and/or have work performed on it. If heated, the entropy of the system would increase; if reversible work were to be done, the entropy would not increase.
 
  • #5
I see...so does that make it unreasonable to speak of the equilibrium states of an isolated system?
 
  • #6
Gear300 said:
I see...so does that make it unreasonable to speak of the equilibrium states of an isolated system?

No, but it seems unreasonable to speak of the temperature of a system whose energy could not be altered, even in theory.
 
  • #7
Gear300 said:
∂σ/∂U = 1/g(∂g/∂U), in which g is the multiplicity of states. Let us say that the system is uncoupled, then wouldn't the equilibrium condition imply ∂g/∂U = 0 (or is that temperature is defined specifically for a coupling between systems)?

Temperature is defined by the zeroth law of thermodynamics, i.e. transitivity of equilibrium: If two systems A and B are in equilibriumwhen brought into contact and B and C are also in equilibrium when brought into contact, then A and C will also be in equilibrium when brought into contact. This allows the introduction of temperature specifically for coupled systems.
 
  • #8
Gear300 said:
I see...so does that make it unreasonable to speak of the equilibrium states of an isolated system?

Yes, temperature is defined in terms of systems being coupled. If two systems are thermally connected and allowed to equilibrate, their temperatures are, by the zeroth law of thermodynamics, the same. An isolated system that can never be connected to another system, can never be connected to a thermometer, and so its temperature will remain unknown, but that's a technological problem. It still has a temperature, it just cannot be practically measured.
 

1. What is temperature?

Temperature is a measure of the average kinetic energy of the particles in a substance. It is typically measured in degrees Celsius (°C) or Fahrenheit (°F), and is directly related to the rate of particle movement. The higher the temperature, the faster the particles move and the more energy they possess.

2. How is temperature related to entropy?

Temperature and entropy are closely related, as they both describe the state of a system. Generally, as temperature increases, so does the entropy of a system. This is because the increased kinetic energy of the particles allows for more possible arrangements, resulting in a higher level of disorder.

3. Can temperature and entropy be measured independently?

While temperature and entropy are related, they can be measured independently. Temperature can be measured using a thermometer, while entropy is typically calculated using statistical mechanics and thermodynamics principles. However, changes in temperature can be used to infer changes in entropy.

4. What is the relationship between temperature and phase changes?

Temperature plays a crucial role in phase changes, such as melting and boiling. This is because at a certain temperature, the particles in a substance have enough energy to overcome the intermolecular forces holding them together and change from one phase to another. This is known as the substance's melting or boiling point.

5. How does temperature affect the transfer of heat?

Temperature is a key factor in the transfer of heat, as heat always flows from areas of high temperature to areas of low temperature. The larger the difference in temperature between two objects, the faster heat will transfer between them. This is why a hot cup of coffee will cool down faster in a cold room compared to a warm room.

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