Van der Waals Equation and specific heat

[Kelsi_Jade]

The problem reads:
Consider a gas with constant specific heat c_v and the van der waals equation of state
(P+ a/v²)(v-b)=RT , where v=V/n

A) Find du and the specific internal energy u=U/n
B) Find ds and the specifi entropy s=S/n


Here's what I've tried so far:

A) I took the initial equation and subbed in the v=V/n :
(P+ a/(V/n)²)(V/n-b)=RT

I know that du= c_vdT which can be rearranged c_v = du/dT so that is where I can find the du is from the specific heat. However, I don't understand how the van der waals equation can be related mathematically to specific heat?
I know that a=a measure of attraction between particles and b=the volume excluded by a mol of particles. Do I need to solve for these first?

Any help is appreciated!

 

[Chestermiller]

The problem reads:
Consider a gas with constant specific heat c_v and the van der waals equation of state
(P+ a/v²)(v-b)=RT , where v=V/n

A) Find du and the specific internal energy u=U/n
B) Find ds and the specifi entropy s=S/n


Here's what I've tried so far:

A) I took the initial equation and subbed in the v=V/n :
(P+ a/(V/n)²)(V/n-b)=RT

I know that du= c_vdT which can be rearranged c_v = du/dT so that is where I can find the du is from the specific heat. However, I don't understand how the van der waals equation can be related mathematically to specific heat?
I know that a=a measure of attraction between particles and b=the volume excluded by a mol of particles. Do I need to solve for these first?

Any help is appreciated!

dU = c_v only for an ideal gas. For a non-ideal gas, u also depends on pressure P. Have you learned the general equation for du in terms of dT and dP?

 

[Kelsi_Jade]

I found this equation:
dU=CvdT+ [T(∂P/∂T)_v - P]dV
but it doesn't mention anything about being related to van der waals' equation so I'm not sure if this is relevant?? Honestly, our text doesn't go much into detail on van der waals so I'm pretty lost..

 

[Chestermiller]

I found this equation:
dU=CvdT+ [T(∂P/∂T)_v - P]dV
but it doesn't mention anything about being related to van der waals' equation so I'm not sure if this is relevant?? Honestly, our text doesn't go much into detail on van der waals so I'm pretty lost..

Your equation is supposed to apply to any gas (or liquid), so it should be applicable to a vdW gas. Yes, the vdW equation is relevant. So you substitute the vdW equation into the term in parenthesis. Incidentally, the dV should be dv.

 

[paranormal]

First, let's rearrange the van der Waals equation to solve for P:
P = (RT/(v-b)) - (a/v^2)

Now, we can use the ideal gas law to relate pressure to temperature and volume:
PV = nRT
P = (nRT)/V

Substituting this into our rearranged van der Waals equation, we get:
(nRT)/V = (RT/(v-b)) - (a/v^2)

Simplifying, we get:
nRT = RT(1/(v-b)) - (a/v^2)

Now, we can rearrange this equation to solve for v:
v = (nRT + a)/(P + (nRT)/V)

Next, we can use the definition of specific internal energy (u = U/n) to find du:
du = cvdT = (nRT + a)/(P + (nRT)/V) * dT

We can also use the definition of specific entropy (s = S/n) to find ds:
ds = cvdT/T = (nRT + a)/(P + (nRT)/V) * dT/T

So, to summarize:
A) du = (nRT + a)/(P + (nRT)/V) * dT
B) ds = (nRT + a)/(P + (nRT)/V) * dT/T

To relate this back to the van der Waals equation, we can see that the specific heat (cv) is related to the term (nRT + a)/(P + (nRT)/V). This term takes into account the attractive forces between particles (a), the excluded volume (b), and the ideal gas behavior (nRT/V). Therefore, the van der Waals equation and specific heat are related through this term, which accounts for the non-ideal behavior of gases.