Question

Does the temperature in the Clausius inequality relation have to be absolute temperature? Why?

Short answer

Answer: The temperature in the Clausius inequality has to be an absolute temperature to represent the true thermodynamic properties of a system and abide by the second law of thermodynamics. Using absolute temperature ensures that heat flow, entropy changes, and other important thermodynamic quantities are accurately represented and eliminates the possibility of negative temperatures causing misinterpretations in the inequality.

Step-by-step solution

Definition of Clausius inequality

The Clausius inequality is a mathematical formulation that arises from the second law of thermodynamics. It states that for any cyclic process, the cyclic integral of the heat supplied to the system divided by the temperature (\(\frac{Q}{T}\)) is always less than or equal to zero, which is mathematically represented as: $$\oint \frac{\delta Q}{T} \leq 0$$ Here, \(\delta Q\) is the infinitesimal amount of heat received by the system, and \(T\) is the temperature.

Absolute temperature

Absolute temperature is a measure of temperature that uses the Kelvin scale, which starts at 0 K (absolute zero). In absolute temperature, the values are always positive, as there is no such thing as negative Kelvin. Absolute temperature better represents the energy content of a system, making it an essential aspect of thermodynamics.

Importance of absolute temperature in Clausius inequality

The reason why the temperature in the Clausius inequality has to be an absolute temperature (in Kelvin) and not a temperature relative to another scale (e.g., Celsius or Fahrenheit) is that absolute temperature reflects the true thermodynamic properties of a system. Using a relative scale with negative values can lead to incorrect conclusions from the Clausius inequality, as \(\frac{Q}{T}\) may yield positive values even when heat flows from a colder body to a warmer one, which is a violation of the second law of thermodynamics. Using absolute temperature ensures that heat flow, entropy changes, and other important thermodynamic quantities are accurately represented. It eliminates the possibility of negative temperatures causing misinterpretations in the inequality. In conclusion, the temperature in the Clausius inequality has to be an absolute temperature to represent the true thermodynamic properties of a system and abide by the second law of thermodynamics.