Which of the following expression is known as Clausius inequality ?

Options

A. `oint(dq)/(T)le 0`
B. `oint(ds)/(T)= 0`
C. `oint(T)/(dq)le 0`
D. `oint(dq)/(T)ge 0`

Answer

To determine which expression is known as the Clausius inequality, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Clausius Inequality**:
The Clausius inequality is a fundamental principle in thermodynamics that relates to the second law of thermodynamics. It applies to closed systems and is used to describe the direction of spontaneous processes.

2. **Identify the Cyclic Process**:
The Clausius inequality is particularly relevant during cyclic processes. A cyclic process is one where the initial and final states of the system are the same. This means that the system returns to its original state after completing the cycle.

3. **Formulate the Clausius Inequality**:
The mathematical expression for the Clausius inequality is given by:
\[ \oint \frac{dq}{T} \leq 0 \]
Here, \( dq \) represents the infinitesimal heat exchanged, and \( T \) is the absolute temperature at which the heat exchange occurs. The symbol \( \oint \) indicates that this is a cyclic integral, meaning it is taken over a closed loop in the thermodynamic process.

4. **Interpret the Inequality**:
The inequality states that the cyclic integral of the heat transfer divided by the temperature is less than or equal to zero. This implies that for any cyclic process, the total entropy change of the system and its surroundings must be greater than or equal to zero, aligning with the second law of thermodynamics.

5. **Conclude the Expression**:
Therefore, the expression known as the Clausius inequality is:
\[ \oint \frac{dq}{T} \leq 0 \]

### Final Answer:
The expression known as the Clausius inequality is:
\[ \oint \frac{dq}{T} \leq 0 \]