Solving for final temp of adiabatic ideal air expansion

[takeachance]

Homework Statement

Given a problem where air expands adiabatically with a given initial temperature, initial pressure, final pressure, and gamma (ratio of specific heats) , how would one go about solving for the final temperature? I'm just eager to get a general sense of direction.

Homework Equations

$PV^\gamma = constant$
(but it doesn't feel particularly useful since volume isn't provided)
$\frac{P_i V_i}{T_i}=\frac{P_f V_f}{T_f}$
(but again, I'm not sure what to do in the absence of volume)

The Attempt at a Solution

Just looking for a sense of direction. I guess that either the volume can be ignored/cancelled or there is an equation that doesn't require volume.

 

[stockzahn]

Homework Statement

Given a problem where air expands adiabatically with a given initial temperature, initial pressure, final pressure, and gamma (ratio of specific heats) , how would one go about solving for the final temperature? I'm just eager to get a general sense of direction.

Homework Equations

$PV^\gamma = constant$
(but it doesn't feel particularly useful since volume isn't provided)
$\frac{P_i V_i}{T_i}=\frac{P_f V_f}{T_f}$
(but again, I'm not sure what to do in the absence of volume)

The Attempt at a Solution

Just looking for a sense of direction. I guess that either the volume can be ignored/cancelled or there is an equation that doesn't require volume.

For an isentropic change of state there is a correlation between $p$ and $T$ depending on $\gamma$ in the form of

$$\left(\frac{p}{T}\right)^{f\left(\gamma\right)} = const.$$

You can find this exponent $f\left(\gamma\right)$ very easily in the internet.

 

[takeachance]

For an isentropic change of state there is a correlation between $p$ and $T$ depending on $\gamma$ in the form of

$$\left(\frac{p}{T}\right)^{f\left(\gamma\right)} = const.$$

You can find this exponent $f\left(\gamma\right)$ very easily in the internet.

Just to double check, $f(γ)$ is perfectly okay as a constant, yes? I guess when I was searching for this I wasn't using the right keywords, because this is my first time seeing P and T related in such a manner in recent memory. Thanks for the help.

 

[stockzahn]

Just to double check, $f(γ)$ is perfectly okay as a constant, yes? I guess when I was searching for this I wasn't using the right keywords, because this is my first time seeing it in recent memory. Thanks for the help.

First of all, I just saw that I made a mistake, sorry. Of course the correlation should be $Tp^{f\left(\gamma\right)} = const$ similar to the correlation you posted

And yes, the function of $\gamma$ is constant for an ideal gas. If you are searching for "isentropic change of state" or "isentropic process" you should find what you are looking for.

 

[takeachance]

First of all, I just saw that I made a mistake, sorry. Of course the correlation should be $Tp^{f\left(\gamma\right)} = const$ similar to the correlation you posted

And yes, the function of $\gamma$ is constant for an ideal gas. If you are searching for "isentropic change of state" or "isentropic process" you should find what you are looking for.

Just a quick search reveals this website that suggests that $TP^{(1-\gamma)/\gamma} = constant$ and not just $TP^{\gamma} = constant$. Any thoughts?

Messing around with some quick numbers seems to suggest that $TP^{\gamma} = constant$ gives an absurdly high constant.

 

[stockzahn]

Just a quick search reveals this website that suggests that $TP^{(1-\gamma)/\gamma} = constant$ and not just $TP^{\gamma} = constant$. Any thoughts?

Just a quick search reveals this website that suggests that $TP^{(1-\gamma)/\gamma} = constant$ and not just $TP^{\gamma} = constant$. Any thoughts?

Messing around with some quick numbers seems to suggest that $TP^{\gamma} = constant$ gives an absurdly high constant.

$f\left(\gamma\right)$ can stand for each function containing $\gamma$.

 

[takeachance]

$f\left(\gamma\right)$ can stand for each function containing $\gamma$. I use the correlation is: $TP^{\gamma/(\gamma-1)} = constant$. Maybe those are equivalent, I didn't check.

Well, I think that get's me to where I need to be. Regardless, all of the online sources I've consulted suggest using $TP^{((γ−1)/γ)}$ or some variation, but this compared to your preferred one seem to yield different exponents.

 

[stockzahn]

$f\left(\gamma\right)$ can stand for each function containing $\gamma$.

Yes, sorry again, your first proposal, according to the website you quoted was right. It's $Tp^{(1-\gamma)/\gamma}$, that's the correlation you can use to solve the problem. You can find these correlation for all "special" changes of states (isochor, isothermal, ...).

Edit: Really sorry for my mistakes, no excuse for that. Next time I concentrate, when responding...

 

[Chestermiller]

What you do is substitute $V=nRT/P$ into your original equation. Then you have an equation involving only P and T.

 

[takeachance]

What you do is substitute $V=nRT/P$ into your original equation. Then you have an equation involving only P and T.

Thanks, I appreciate an alternate approach, but I don't know how much of the gas I have. I don't think I would be able to deal with n. Does your suggestion still work without knowing or having a way to solve for n?

 

[Chestermiller]

Thanks, I appreciate an alternate approach, but I don't know how much of the gas I have. I don't think I would be able to deal with n. Does your suggestion still work without knowing or having a way to solve for n?

n is a constant.

 

[takeachance]

n is a constant.

I thought n was the number of moles of gas you have and R was a gas constant?

Edit: Do you mean that the n is constant in the situation (unchanging during expansion) so the n will eventually not be an issue?

 

[Chestermiller]

Yes. That's exactly what I mean. It will be absorbed into the constant to form a new constant C'