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# Work Done In Adiabatic Process Derivation

An adiabatic process is a thermodynamic process during which no heat energy is transferred across the boundaries of the system. This does not mean the temperature is constant, but rather that no heat is transferred into or out of the system. In this article let us learn about the adiabatic process, examples, adiabatic process equation derivation and the adiabatic index.

• The vertical flow of air in the atmosphere
• When the interstellar gas cloud expands or contracts.
• The turbine is an example of the adiabatic process as it uses heat as a source to produce work.

The equation for an adiabatic process can be derived from the first law of thermodynamics relating to the change in internal energy dU to the work W done by the system and the heat dQ added to it.

dU=dQ-dW

dQ=0 by definition,

Therefore, 0=dQ=dU+dW

The work done dW for the change in volume V by dV is given as PdV.

The first term is related to specific heat which is defined as the heat added per unit temperature change per mole of a substance. The heat that is added increases the internal energy U such that it justifies the definition of specific heat at constant volume given as

$$\begin{array}{l}C_{v}=\frac{dU}{dT}\frac{1}{n}\end{array}$$

Where,

n: number of moles

Therefore,

$$\begin{array}{l}0=nC_{v}dT+PdV..(eq.1)\end{array}$$

From the ideal gas law, we have

nRT=PV (eq.2)

Therefore, nRdT=PdV+VdP (eq.3)

By combining equation 1 and equation 2, we get

$$\begin{array}{l}-PdV=nC_{v}dT=\frac{C_{v}}{R}(PdV+VdP)\end{array}$$
$$\begin{array}{l}0=(1+\frac{C_{v}}{R})PdV+\frac{C_{v}}{R}VdP\end{array}$$
$$\begin{array}{l}0=\frac{R+C_{v}}{C_{v}}(\frac{dV}{V})+\frac{dP}{P}\end{array}$$

When the heat is added at constant pressure Cp, we have

$$\begin{array}{l}C_{p}=C_{v}+R\end{array}$$
$$\begin{array}{l}0=\gamma (\frac{dV}{V})+\frac{dP}{P}\end{array}$$

Where the specific heat ɣ is given as:

$$\begin{array}{l}\gamma\equiv \frac{C_{p}}{C_{v}}\end{array}$$

From calculus, we have,

$$\begin{array}{l}d(lnx)=\frac{dx}{x}\end{array}$$
$$\begin{array}{l}0=\gamma d(lnV)+d(lnP)\end{array}$$
$$\begin{array}{l}0=d(\gamma lnV+lnP)=d(lnPV^{\gamma })\end{array}$$
$$\begin{array}{l}PV^{\gamma }=constant\end{array}$$

Hence, the equation is true for an adiabatic process in an ideal gas.
Learn the adiabatic process equation derivation in detail and also know how the adiabatic process is achieved in the below video. The adiabatic index is also known as the heat capacity ratio and is defined as the ratio of heat capacity at constant pressure Cp to heat capacity at constant volume Cv. It is also known as the isentropic expansion factor and is denoted by ɣ.

$$\begin{array}{l}\gamma =\frac{C_{p}}{C_{v}}=\frac{c_{p}}{c_{v}}\end{array}$$

Where,

C: heat capacity

c: specific heat capacity

The adiabatic index finds application reversible thermodynamic process involving ideal gases and speed of sound is also dependent on the adiabatic index.

## Frequently Asked Questions – FAQs

True.

### What is meant by the adiabatic index?

The adiabatic index is the ratio of heat capacity at constant pressure Cp to heat capacity at constant volume Cv.
$$\begin{array}{l}\gamma =\frac{C_{p}}{C_{v}}=\frac{c_{p}}{c_{v}}\end{array}$$

### Give an example of an adiabatic process.

Example of adiabatic process: The turbine uses heat to produce work.

### Give the expression for the adiabatic process in an ideal gas.

$$\begin{array}{l}PV^{\gamma }=constant\end{array}$$

### The adiabatic index is also known as?

The adiabatic index is also known as the heat capacity ratio.