# Polytropic Process for an Ideal Gas

• LunaFly
In summary, a polytropic process is one in which PV\gamma = constant. Assuming n is constant, this does not mean that the temperature remains constant. In fact, in most cases, the temperature will change during a polytropic process. This is because PV\gamma is equal to nrT only when \gamma = 1. In other cases, there is no simple relationship between PV\gamma and T. Therefore, a polytropic process does not imply that nRT is a constant, even when n is constant.

#### LunaFly

Hi,

I have a general question about polytropic processes when working with an ideal gas.

A polytropic process is one for which PV$\gamma$ = constant.

What I am wondering is if we assume that n is constant, does that mean that the temperature does not change for the process?

I see that PV$\gamma$ = nrT = constant, but for some reason it doesn't seem right that a polytropic process would be equivalent to an isothermal process if the number of moles of the system doesn't change.

A little light on the subject would be great.

Thanks,

-Luna

LunaFly said:
Hi,

I have a general question about polytropic processes when working with an ideal gas.

A polytropic process is one for which PV$\gamma$ = constant.

What I am wondering is if we assume that n is constant, does that mean that the temperature does not change for the process?
No. A good example of such a process is adiabatic expansion/compression, for which ##T## changes.

LunaFly said:
I see that PV$\gamma$ = nrT
That equation is only valid if ##\gamma = 1##. Otherwise, there is no simple relationship between ##PV^\gamma## and ##T##.

Ok this concept finally sunk in.

PV$\gamma$ does not equal nRT unless $\gamma$=1. So a polytropic process in no way implies that nRT is a constant (even when n is constant).

DrClaude, thanks for the input!

LunaFly said:
Ok this concept finally sunk in.

PV$\gamma$ does not equal nRT unless $\gamma$=1. So a polytropic process in no way implies that nRT is a constant (even when n is constant).

Consider a process for an ideal gas where ##PV^\gamma = \textrm{const.}##. Using the ideal gas law, ##PV=nRT##, we have
\begin{align} P_iV_i^\gamma &= P_fV_f^\gamma \\ \left( \frac{n R T_i}{V_i} \right) V_i^\gamma &= \left( \frac{n R T_f}{V_f} \right) V_f^\gamma \\ T_i V_i^{\gamma - 1} &= T_f V_f^{\gamma - 1} \end{align}
or ##TV^{\gamma - 1} = \textrm{const.}##
Obviously, if ##V## changes during the process, then ##T## changes (except if ##\gamma = 1##).

1 person

## 1. What is a polytropic process?

A polytropic process is a type of thermodynamic process in which a gas undergoes a change in volume while its pressure remains constant. This process is often used to model the expansion or compression of gases in real-life situations.

## 2. What is an ideal gas?

An ideal gas is a theoretical gas that follows the gas laws perfectly, meaning it has no intermolecular forces and its particles have no volume. This allows for simpler calculations and is often used in thermodynamic analyses.

## 3. How is the polytropic process represented mathematically?

The polytropic process can be represented by the equation P1V1^n = P2V2^n, where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and n is the polytropic index, which represents the relationship between pressure and volume.

## 4. What is the significance of the polytropic index in a polytropic process?

The polytropic index, also known as the exponent of the process, determines the type of process that is occurring. If n = 1, the process is isothermal, if n = 0, the process is isobaric, and if n = γ = c_p/c_v (ratio of specific heats), the process is adiabatic.

## 5. What are some real-life applications of the polytropic process?

The polytropic process can be observed in various real-life situations, such as the expansion and compression of gases in internal combustion engines, the compression and expansion of air in refrigerators and air conditioners, and the compression of air in scuba tanks.