## Carnot Cycle – Processes

In a** Carnot cycle**, the system executing the cycle undergoes a series of** four internally reversible processes**: **two isentropic processes** (reversible adiabatic) alternated with **two** **isothermal processes**:

**isentropic compression**– The gas is compressed adiabatically from state 1 to state 2, where the temperature is**T**. The surroundings do work on the gas, increasing its internal energy and compressing it. On the other hand the_{H}**entropy**remains**unchanged**.**Isothermal expansion**– The system is placed in contact with the reservoir at**T**. The gas expands isothermally while receiving energy Q_{H}_{H}from the hot reservoir by heat transfer. The temperature of the gas does not change during the process. The gas does work on the surroundings. The total entropy change is given by:*∆S = S*_{1}*– S*_{4}*= Q*_{H}*/T*_{H}**isentropic expansion**– The gas expands adiabatically from state 3 to state 4, where the temperature is**T**. The gas does work on the surroundings and loses an amount of internal energy equal to the work that leaves the system. Again the entropy remains unchanged._{C}**isothermal compression**– The system is placed in contact with the reservoir at**T**. The gas compresses isothermally to its initial state while it discharges energy Q_{C}_{C}to the cold reservoir by heat transfer. In this process the surroundings do work on the gas. The total entropy change is given by:*∆S = S*_{3}*– S*_{2}*= Q*_{C}*/T*_{C}

## Isentropic Process

An **isentropic process** is a** thermodynamic process**, in which the **entropy** of the fluid or gas remains constant. It means the **isentropic process** is a special case of an **adiabatic process** in which there is no transfer of heat or matter. It is a **reversible adiabatic process**. The assumption of no heat transfer is very important, since we can use the adiabatic approximation only in **very rapid processes**.

**Isentropic Process and the First Law**

For a closed system, we can write the **first law of thermodynamics in terms of enthalpy**:

**dH = dQ + Vdp**

**or**

**dH = TdS + Vdp**

**Isentropic process (dQ = 0):**

**dH = Vdp → W = H**_{2}** – H**_{1}** → H**_{2}** – H**_{1}** = C**

_{p}

*(T*

_{2}

*– T*

_{1}

*)**(for ideal gas)*

**Isentropic Process of the Ideal Gas**

The **isentropic process** (a special case of adiabatic process) can be expressed with the **ideal gas law** as:

*pV ^{κ} = constant*

or

**p _{1}V_{1}^{κ} = p_{2}V_{2}^{κ}**

in which **κ = c _{p}/c_{v}** is the ratio of the

**specific heats**(or

**heat capacities**) for the gas. One for

**constant pressure (c**

_{p}**)**and one for

**constant volume (c**

_{v}**)**. Note that, this ratio

**κ**

**= c**is a factor in determining the speed of sound in a gas and other adiabatic processes.

_{p}/c_{v}## Isothermal Process

An **isothermal process** is a **thermodynamic process**, in which the **temperature** of the system **remains constant** (T = const). The heat transfer into or out of the system typically must happen at such a slow rate in order to continually adjust to the temperature of the reservoir through heat exchange. In each of these states the **thermal equilibrium** is maintained.

**Isothermal Process and the First Law**

The classical form of the first law of thermodynamics is the following equation:

**dU = dQ – dW**

In this equation dW is equal to **dW = pdV** and is known as the boundary work.

In **isothermal process** and the **ideal gas**, all heat added to the system will be used to do work:

**Isothermal process (dU = 0):**

**dU = 0 = Q – W → W = Q **

*(for ideal gas)*

**Isothermal Process of the Ideal Gas**

The **isothermal process **can be expressed with the **ideal gas law** as:

*pV = constant*

or

**p _{1}V_{1} = p_{2}V_{2}**

On a p-V diagram, the process occurs along a line (called an isotherm) that has the equation** p = constant / V**.

See also: Boyle-Mariotte Law

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