[TA1LGUNN3R]

Doing some homework, not sure if the answers are believable, and I want to make sure I'm doing this right.

At 98 °F, water exists as a gas and a liquid with a vapor pressure of 47 torr.  a) Calculate the number density and molar density of gaseous water; and b) calculate the packing fraction of gas phase water.  The liquid phase has η≈0.4.  Assume that a water molecule is spherical with a diameter of 3.0Å  Packing fraction: η≡(volume per particle)*number of particles per unit volume.

So obviously I first converted the units.  No problem there.  Then I used ρ=PM/RT and n=N_A*ρ/M, which gives n=(N_A*P)/RT.  Sub in the values and I get 1.46E21 L^-1.  This should be number density, yes?  The number of particles per unit volume. 

Then to packing fraction.  η=(4π/3)*(1.5E-11 m)³*(1.46E24 m^-3), which yields 2.E-8.  It is dimensionless, but is the packing fraction of gas really that much smaller than the liquid?

Thanks in advance,
TG

[mjc123]

Slight mistake. 1Å = 10^-10m, so η = 2e-5.
A rough rule of thumb for low-MW substances is that the volume of the gas at 1 atm is roughly 1000 times that of the liquid. In this case your gas is at much less than 1 atm.
Number density is right, what about the molar density?

[TA1LGUNN3R]

Ah okay, I'll fix the unit conversion.  Molar density wouldn't I just divide the number density by Avogadro's constant?

[TA1LGUNN3R]

Ok I've been working on this one for a while now.

Show that

[tex]\alpha_P=-(\frac {\partial ln \rho} {\partial T})_P[/tex]

[tex]\kappa_T=(\frac {\partial ln \rho} {\partial T})_T[/tex]

Just a hint would help.  I've tried subbing, implicit partial differentiation, but can't figure where ln ρ fits in.  I've gotten as far as implicitly differentiating the ideal gas law to get the thermal expansion coefficient.

α_P being thermal expansion coefficient, κ_T being isothermal compressibility, and ρ=n/V.

Thanks in advance.

[mjc123]

This is a change of subject, you should really start a new thread.
Would it help to say d(lnρ)/dT = 1/ρ*dρ/dT?
Should your second equation read κ_T = (dlnρ/dP)_T?

[TA1LGUNN3R]

Sorry, I didn't want to unnecessarily clutter up the forum, since I figure I'll be posting a bit as needed.   

Yes, it should be dP in the kappa equation, I mis-typed.

Hm, that didn't really help.  Working from the right hand side of α_P, I get:

[tex]-(\frac {\partial}{\partial T}*ln \rho)=-(\frac {1}{\rho}*\frac {\partial \rho}{\partial T})=-(\frac {V}{n}*n\frac {(\partial \frac {1}{V})}{\partial T})[/tex]

After I nix the n, I'm not sure what to do.  Is there a particular identity I'm forgetting here?

[mjc123]

what is d(1/V)/dT?
And what is the definition of α?

[TA1LGUNN3R]

Ah, okay.  d(1/V)/dT become -(1/V²)*(dV/dT).

Thank you.  It's been a couple of years since I actually took calc, so sometimes my skill are a bit rusty.

[Mezuzah]

Where did you find the equation: "n=N_A*ρ/M" other than wikipedia. This is the exact same question (work for word, value for value) from our classes homework that my instructor handed out. That equation isn't in lecture or the textbook.

Were you able to derive this or did you get it from the solutions manual and/or wikipedia or...?

Mezuzah

[TA1LGUNN3R]

Where did you find the equation: "n=N_A*ρ/M" other than wikipedia. This is the exact same question (work for word, value for value) from our classes homework that my instructor handed out. That equation isn't in lecture or the textbook.

Were you able to derive this or did you get it from the solutions manual and/or wikipedia or...?

Mezuzah

Honestly, I wiki'd it.  But you can figure most equations if you know the general gist and units needed.  It stands to reason that number density would be the number of atoms per unit of volume (N/L), and the book gives you the formula for mass density (ρ=PM/RT).  You know that mass density is g/L, so you would have to multiply by N/g to yield N/L.  Avogadro's constant is a unit of N/mol, so if you divided A by molecular mass, you'd get N/g.