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 need to understand the concept behind it. The Clausius inequality is a fundamental principle in thermodynamics that relates to the second law of thermodynamics. Here’s a step-by-step solution:

### Step 1: Understand the Clausius Statement
The Clausius statement of the second law of thermodynamics states that it is impossible to construct a device that operates in a cyclic process and produces no other effect than the transfer of heat from a colder body to a hotter body.

### Step 2: Identify the Mathematical Expression
The Clausius inequality can be mathematically expressed as:
\[ \Delta S \geq \frac{Q}{T} \]
where:
- \(\Delta S\) is the change in entropy,
- \(Q\) is the heat exchanged,
- \(T\) is the absolute temperature of the system.

### Step 3: Interpret the Inequality
This expression indicates that the change in entropy of a system is always greater than or equal to the heat transferred divided by the temperature. This means that in any real process, the entropy of the universe tends to increase, which aligns with the second law of thermodynamics.

### Step 4: Conclusion
Thus, the expression \(\Delta S \geq \frac{Q}{T}\) is known as the Clausius inequality. It emphasizes the impossibility of transferring heat from a colder body to a hotter body without some form of work input or without increasing the total entropy of the system.

### Summary
The Clausius inequality is represented by the expression:
\[ \Delta S \geq \frac{Q}{T} \]

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