[mana]

hi all
in the following graph, as you know we can see the deviation of real gas from an ideal gas, my question is: in extremely high pressures where the free volume (the space between gas particles) decreases, why keeping the container volume in the numerator leads the PV/RT values more than ideal gas? is it because of gas pressure increased?
I mean in this situation we keep the container volume at that fraction ( and it is equal to the volume of an ideal gas) but we know the exact value is less but herein due to pressure increasing the fraction PV/RT rises above the ideal gas. am I right?

[mjc123]

What do you mean by "due to pressure increasing"? You have a graph of PV/RT vs. pressure. Do you mean, for a given point, at that pressure the pressure increases? That is meaningless. Do you mean that the effective pressure is greater than the measured pressure due to the intermolecular forces - the +a/V² term in the vdW equation? The graph tells you that in the region of interest the effect of particle volume (the -b term) predominates, so that can't be the explanation.

The misunderstanding seems to lie in the sentence "in this situation we keep the container volume at that fraction ( and it is equal to the volume of an ideal gas)". A vdW gas behaves like an ideal gas whose pressure is P + a/V² and whose volume is V - b. The measured (container) volume is greater than the corresponding ideal gas volume, which is why PV > RT.

[mana]

What do you mean by "due to pressure increasing"?

it is a little difficult for me to explain, this explanation is from General Chemistry, 8th edition, Martin Silberberg page 243:
"2. Effect of particle volume. At normal Pext, the space between particles (free
volume) is enormous compared with the volume of the particles themselves (particle
volume); thus, the free volume is essentially equal to V, the container volume in PV/RT.
At moderately high Pext and as free volume decreases, the particle volume makes up
an increasing proportion of the container volume (Figure 5.25). At extremely high
pressures, the space taken up by the particles themselves makes the free volume
significantly less than the container volume. Nevertheless, we continue to use the
container volume for V in PV/RT, which causes the numerator, and thus the ratio, to
become artificially high. This particle volume effect increases as Pext increases, eventually
outweighing the effect of interparticle attractions and causing PV/RT to rise
above the ideal value."
I think it means that we assume the volume for ideal gas and real gas the same, and now I am wondering if the volumes are the same so why do the curves rise above the ideal gas line where PV/RT=1, I'm not sure but I guess although we assume the same value for the volumes, but in real gas pressure must be higher due to reduced volume (which we neglect in PV/RT)

[mana]

I hope I could explain my question exactly, and sorry for my poor English

[Corribus]

You might read around about the compressibility factor; https://en.wikipedia.org/wiki/Compressibility_factor

[mjc123]

I think it means that we assume the volume for ideal gas and real gas the same

No it does not, quite the reverse. The paragraph you quote explains why the true volume (equivalent to the ideal gas volume) is lower than the measured (container) volume. We continue to use the container volume V (as well as the measured pressure P) because these are easily measurable quantities, which the free volume (equivalent to ideal gas volume) is not. But then we find that PV/RT > 1 because V > V_ideal.

[mana]

I think it means that we assume the volume for ideal gas and real gas the same

But then we find that PV/RT > 1 because V > V_ideal.

V _ideal mustn't be more than V ?  do you mean V_real by V?

[mjc123]

By V I mean V as used in the graph, i.e. the container volume, the actual volume you measure.
By V_ideal I mean the free volume, the space that the molecules have to move about in; what is represented by (V-b) in the vdW equation, and in that approximation is assumed to behave like the volume of an ideal gas.
I'm not sure what you mean by V_real.

To sum up: in the regime where "effect of particle volume predominates", we are approximating
P(V-b) = RT
hence PV = RT + Pb
PV > RT

[mana]

I'm not sure what you mean by V_real.

I thought V_ideal means container volume and V (or V_real) means V-b,

thanks for your help

[mjc123]

V can't mean V-b, or you're really in trouble!
But irrespective of what notation we use, do you understand the principle physically?

[mana]

V can't mean V-b, or you're really in trouble!
But irrespective of what notation we use, do you understand the principle physically?

I know by reducing volume the pressure increases and vice versa, but I am not sure that is all I should know in physically principle