Polytropic Process for an Ideal Gas

[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

 

[DrClaude]

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.

I see that PV^\gamma = nrT

That equation is only valid if $\gamma = 1$. Otherwise, there is no simple relationship between $PV^\gamma$ and $T$.

 

[LunaFly]

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!

 

[DrClaude]

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).

Another thing to consider that might be helpful for your understanding:

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 $\gamma = 1$).