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 $\gamma = 1$).

 

[PJC01]

Hello Luna,

Thank you for your question about polytropic processes for an ideal gas. To answer your question, assuming that n (number of moles) is constant does not necessarily mean that the temperature will not change during the process. This is because in a polytropic process, the value of \gamma can vary depending on the conditions of the process. \gamma is known as the polytropic index and it represents the relationship between pressure and volume during the process. In an isothermal process, \gamma is equal to 1, but in a polytropic process, it can have any value between 0 and \infty.

Therefore, even if n is constant, if the polytropic index \gamma is not equal to 1, then the temperature will change during the process. This is because the relationship between pressure and volume is not the same as an isothermal process, where \gamma is equal to 1 and therefore, the temperature remains constant.

I hope this helps clarify your understanding of polytropic processes for an ideal gas. Please let me know if you have any further questions.

Best,