Calculate this value of 1 mole at STP with C_V

[grandpa2390]

Homework Statement

Given $(\frac{∂H}{∂U})_P = (\frac{C_V+ (π_T + P)*V*a}{C_V + V*a*π_T})$
Calculate this value of 1 mole of ideal gas at STP that has constant heat capacity of 12.5 $\frac{J*K}{mol}$

n=1
T = 273.15 K
P = 1 atm
$C_V = 12.5 \frac{J*K}{mol}$
a = ?
$π_T = ?$

Homework Equations

$a = \frac{1}{V}*(\frac{∂V}{∂T})_p$
$π_T = T(\frac{∂P}{∂T})_v - P$

The Attempt at a Solution

So, since this is an ideal gas. I tried to find a by solving PV=nRT for V and then taking the partial with respect to T. I got nR/P
I plugged in the numbers and for a I got .003661

For π_T I did the reverse. I solved for P and then took the partial with respect for V.
I plugged in the values and and got 0.

Plugging these numbers into the given formula above. I keep coming up with 1.00656
The answer is 1.67...
What am I doing wrong?

 

[DrClaude]

Try to go as far as possible algebraically before plugging in numbers. In other words, find analytical expressions for a and π_T for an ideal gas and substitute the values in the equation.

 

[grandpa2390]

no cigar

I got to this after two separate attempts. If I am making an error, it is the same error over and over.

$\frac{C_v*n^2*R^2*T}{PVC_v+n^2*R^2*T-PVnR}$

and when I plug in the numbers, I get .08205
apparently everything should cancel except one R.

I don't know.

 

[DrClaude]

What do you get for a and π_T individually?

 

[grandpa2390]

What do you get for a and π_T individually?

I'm sure that is where I went wrong. either that or i calculated the volume incorrectly.
I got roughly 22 for volume by solving PV=nRT for V

for a, I did v=nRT/P and differentiated. so $\frac{nR}{PV}$ .0037

for pi I got $\frac{nRT}{V} - P$ 0

assuming V is correct...

 

[DrClaude]

I'm sure that is where I went wrong. either that or i calculated the volume incorrectly.
I got roughly 22 for volume by solving PV=nRT for V

Hint: you don't need the volume if you do the algebra to the end.

for a, I did v=nRT/P and differentiated. so $\frac{nR}{PV}$ .0037

for pi I got $\frac{nRT}{V} - P$ 0

Yes, $\pi_T = 0$ and $a = nR / PV$. What do you get then for $(\pi_T + P ) V a$ and for $V a \pi_T$?

 

[grandpa2390]

Hint: you don't need the volume if you do the algebra to the end.Yes, $\pi_T = 0$ and $a = nR / PV$. What do you get then for $(\pi_T + P ) V a$ and for $V a \pi_T$?

gah I did it 3 times already and I keep getting the same result. I'll try a fourth time...
is my answer in post 3 wrong then you say?

 

[DrClaude]

is my answer in post 3 wrong then you say?

Yes. The actual equation is much simpler.

If you can show your derivation, it will be easier for me to help you figure out where it goes wrong.

 

[grandpa2390]

Yes. The actual equation is much simpler.

If you can show your derivation, it will be easier for me to help you figure out where it goes wrong.

that's what I was going to do. but typing it in LaTeX takes so long. if I were to do it in Mathematica, could I just copy and paste? is that possible?

 

[DrClaude]

that's what I was going to do. but typing it in LaTeX takes so long. if I were to do it in Mathematica, could I just copy and paste? is that possible?

Anything that's readable will be fine.

 

[grandpa2390]

Anything that's readable will be fine.

then I can try uploading a scan of me handwriting it? I know it is against the rules, but I'll do it again extra slow and if it is illegible I can just scrap it and do it through mathematica.

 

[DrClaude]

then I can try uploading a scan of me handwriting it? I know it is against the rules, but I'll do it again extra slow and if it is illegible I can just scrap it and do it through mathematica.

We don't like scans in OPs, but we understand it can be hard to recopy everything. Here it will be fine.

 

[grandpa2390]

We don't like scans in OPs, but we understand it can be hard to recopy everything. Here it will be fine.

thank you :)

I'm going to resolve it from scratch. who knows. maybe this time, just because I asked a question, I'll get the right answer. That's usually what happens. I get the wrong answer until I ask someone for help.

 

[grandpa2390]

part 1 I edited the steps so you could easily point out my mistake ;)

 

[grandpa2390]

part 2edit: this page isn't right. I accidentally multiplied by Cv in the denominator

 

[grandpa2390]

I just had a brainstorm (or Eureka! moment)

I could replace V with nRT/P :)

 

[grandpa2390]

Correcting the accidental error in this work in the denominator got me 1.00029

replacing v with nRT/P got me the same thing. :(

 

[DrClaude]

In step 3, simplify to $\pi_T = 0$, as discussed above.

 

[grandpa2390]

In step 3, simplify to $\pi_T = 0$, as discussed above.

$\frac{C_v+nR}{C_v}$ ?

= 1.0065...

 

[grandpa2390]

but why?

shouldn't they be the same?

I get why it simplifies to 0. I didn't see it before but nRT/v = p and P-P = 0

but still... where did I make a mistake. it was definitely easy to in the mess I made

 

[grandpa2390]

disregard this

 

[grandpa2390]

well it is probably going to be really hard if my mistake isn't near the beginning because I messed up the denominator. here is an abbreviated version that is correct (except for it not being correct)

 

[DrClaude]

$\frac{C_v+nR}{C_v}$ ?

= 1.0065...

That's the correct algebraic equation. I don't understand how you get that numerical value. What value if R are you using?