In the context of thermodynamics, a cyclic integral refers to the integral of a property or a process quantity over a complete thermodynamic cycle. For a process quantity like heat ($\delta Q$) or work ($\delta W$), a cyclic integral represents the net amount transferred during the cycle. For a state property (like internal energy, $U$, or entropy, $S$), the cyclic integral is always zero, as the system returns to its initial state.
However, the term "cyclic integral" is famously associated with the cyclic integral of heat divided by temperature, which is fundamental to the second law of thermodynamics and the concept of entropy.
Understanding the Cyclic Integral of $\delta Q / T$
This specific cyclic integral, often written as $\oint \frac{\delta Q}{T}$, is crucial in analyzing thermodynamic cycles. It involves integrating the differential amount of heat transferred ($\delta Q$) at each point in the cycle, divided by the absolute temperature ($T$) at which the transfer occurs.
Based on thermodynamic principles:
- Any heat flow to or from a system can be considered to consist of differential amounts of heat ($\delta Q$).
- The cyclic integral of $\frac{\delta Q}{T}$ can be viewed as the sum of all these differential amounts of heat divided by the temperature at the boundary where the heat transfer takes place.
This integral is key to Clausius's inequality, which states that for any cyclic process:
$\oint \frac{\delta Q}{T} \le 0$
Where the equality holds for reversible cycles and the inequality holds for irreversible cycles.
Significance in Thermodynamics
The cyclic integral of $\delta Q / T$ has profound implications:
- Entropy Change: It directly relates to the concept of entropy ($S$), a state property. For a reversible process between two states 1 and 2, the change in entropy is $\Delta S = \int_1^2 \frac{\delta Q}{T}$.
- Feasibility of Cycles: Clausius's inequality, derived from the cyclic integral, provides a criterion for the feasibility of a thermodynamic cycle. No cycle can have $\oint \frac{\delta Q}{T} > 0$.
- Definition of Absolute Temperature: The concept underpins the thermodynamic definition of absolute temperature scales.
Example
Consider a system undergoing a cycle involving heat exchanges at different temperatures. To calculate the cyclic integral $\oint \frac{\delta Q}{T}$, you would sum (integrate) $\frac{\delta Q}{T}$ for every heat transfer step throughout the cycle.
- If heat $\delta Q_1$ is added at temperature $T_1$, it contributes $\frac{\delta Q_1}{T_1}$.
- If heat $\delta Q_2$ is removed at temperature $T_2$, it contributes $\frac{\delta Q_2}{T_2}$ (where $\delta Q_2$ is negative as heat is removed).
The cyclic integral is the sum of all such contributions over the entire cycle.
In summary, while a cyclic integral generally means an integral over a closed loop in the state space of a system, in thermodynamics, it most commonly refers to the cyclic integral of $\delta Q / T$, which quantifies the sum of differential heat transfers divided by their respective boundary temperatures over a cycle and is fundamental to the second law and entropy.
