# Why 'vd ∝ d' in Pv = nRT?

## Homework Statement

Why 'vd ∝ d' in Pv = nRT? Should it be 'vd ∝ d⁄2'? (where vd = vapour density)

## The Attempt at a Solution

As we know Pv = nRT, PM = dRT, PM / 2 = dRT / 2, M / 2 = d / 2, vd ∝ d (as given by the book) but my question is it should be 'vd ∝ d / 2' as denominator 2 is remeaning.

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phyzguy
Do you understand the meaning of the ∝ symbol ("proportional to")?

• baldbrain
Borek
Mentor
And what is d? Looks to me like it is vapor density as well, so you are proving that something is directly proportional to itself.

• baldbrain
And what is d? Looks to me like it is vapor density as well, so you are proving that something is directly proportional to itself.
d stands for density

Do you understand the meaning of the ∝ symbol ("proportional to")?
When quantities have the same relative size. In other words, they have the same ratio.
Example:
A rope's length and weight are in proportion. When 20m of rope weighs 1kg, then:
• 40m of that rope weighs 2kg
• 200m of that rope weighs 10kg
etc.

Borek
Mentor
d stands for density
Density of what? Sea water? Gold? Vapor of neodymium?

phyzguy
When quantities have the same relative size. In other words, they have the same ratio.
Example:
A rope's length and weight are in proportion. When 20m of rope weighs 1kg, then:
• 40m of that rope weighs 2kg
• 200m of that rope weighs 10kg
etc.
So if X is proportional to Y, then isn't X also proportional to Y/2? or 47*Y? or Y times any constant?

Density of what? Sea water? Gold? Vapor of neodymium?
Density of the gas.

So if X is proportional to Y, then isn't X also proportional to Y/2? or 47*Y? or Y times any constant?
Yes. Then we should write 'vd ∝ d / 2' but the book shows ''vd ∝ d'? That's my question.

Borek
Mentor
Density of the gas.
And how is the density of the gas different from the density of the vapor? Do you understand it is the same thing and saying vd ∝ d makes the same sense as saying 1 ∝ 1?

Then we should write 'vd ∝ d / 2' but the book shows ''vd ∝ d'?
Apparently you have not understood a word of what @phyzguy wrote :(

If X is proportional to Y times any constant, it actually doesn't matter if Y is multiplied or divided by anything - the proportionality still holds, just the constant changes.

And how is the density of the gas different from the density of the vapor? Do you understand it is the same thing and saying vd ∝ d makes the same sense as saying 1 ∝ 1?
I think there is no difference between vapor and gas. So their densities should be the same.

Borek said:
Apparently you have not understood a word of what @phyzguy wrote :(
If X is proportional to Y times any constant, it actually doesn't matter if Y is multiplied or divided by anything - the proportionality still holds, just the constant changes.
It would be very helpful if you provide an example with respect to your explanation above.

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phyzguy
It would be very helpful if you provide an example with respect to your explanation above.
X ∝ Y means that we can write X = λ * Y, where λ is some constant. So the expression vd ∝ d and vd ∝ d/2 mean the same thing. Only the value of the proportionality constant λ is different. Because of this, we typically don't include any constant when writing a proportionality. Since vd ∝ d, vd ∝ d/2, and vd ∝ 13.78234 * d all mean the same thing, we typically just write the simplest expression.

X ∝ Y means that we can write X = λ * Y, where λ is some constant. So the expression vd ∝ d and vd ∝ d/2 mean the same thing. Only the value of the proportionality constant λ is different. Because of this, we typically don't include any constant when writing a proportionality. Since vd ∝ d, vd ∝ d/2, and vd ∝ 13.78234 * d all mean the same thing, we typically just write the simplest expression.
Then can I write vd ∝ 1/2. d (where 1/2 is a constant)?

Lord Jestocost
Gold Member
Lord Jestocost
Gold Member
Then can I write vd ∝ 1/2. d (where 1/2 is a constant)?
That makes no sense. Given the two variables vd and d, vd is directly proportional to d if there is a non-zero constant k such that
vd = k * d

Such a relation is often denoted, by using the symbol ∝, as

vdd

( https://en.wikipedia.org/wiki/Proportionality_(mathematics) )

Mark44
Mentor
Then can I write vd ∝ 1/2. d ?
Yes, but why would you do this? If vd ∝ d, then any of the following would also be correct (but not very useful).

vd ∝ 7d
vd ∝ πd
vd ∝ (1/3)d
etc.
(where 1/2 is a constant)
It's obvious that 1/2 is a constant -- you don't need to say this.

I think there is no difference between vapor and gas
You need to clear your basics then the ideal gas equation.

How on earth is 1/2 of M = 1/2 of d.It is not even dimensionally correct.

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Borek
Mentor
I think there is no difference between vapor and gas.
You need to clear your basics then the ideal gas equation.
Technically OP is correct, we refer to gas above liquid as vapor, but they are in no way different from each other.

A gas refers to a substance that has a single defined thermodynamic state at room temperature whereas a vapor refers to a substance that is a mixture of two phases at room temperature, namely gaseous and liquid phase.
From differencebetween.net

V in subscript d as the OP is talking about, is Vapour density. Say vapor density of air, we need to calculate each gase's molecular weight. Nitrogen 0.7 mole x 28 + 0.21 x 32 for Oxygen + 0.09 x mass of other gas. While gas density refers simply to mass/volume of a gas say nitrogen only.

Vapour density of air makes sense because air is a homogeneous mixture of different gases in different proportions. But Vapor density of Oxygen technically refers to its gas density. (You are right here). Monoatomic elements have no vapour density.

Vapour Density is always measured relative to Hydrogen

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Borek
Mentor
A gas refers to a substance that has a single defined thermodynamic state at room temperature whereas a vapor refers to a substance that is a mixture of two phases at room temperature, namely gaseous and liquid phase.
The nomenclature is clumsy, the old thread you linked to contains the same problem: several definitions are mixed and the meaning of both words depend on whom you are talking to. Please read it as a whole.

Most common (and most typical for those learning the gas properties) is a problem with the difference between 'vapor' and 'steam' in everyday language (where both steam and vapor can be white and visible because of the droplets present) and technical and science language (where both seem to refer to the gas).

Probably the best approach is to speak about gas as one thing and wet steam as the other, at least it is unambiguous then.

where both steam and vapor can be white and visible because of the droplets present
They are visible due to refraction. And due to presence of an aerosol type mixture.

While steam is water at gas phase, Vapour is a thermodynamic state where two phases can coexist.

The nomenclature is clumsy
Which ones?

Borek
Mentor
Vapour is a thermodynamic state where two phases can coexist.
Wiki quotes General Chemistry textbook as saying

In physics a vapor (American) or vapour (British and Canadian) is a substance in the gas phase at a temperature lower than its critical temperature
which is a different thing than you wrote. That already shows there are problems with the nomenclature, as I assume you are able to quote a source showing your definition.

Which ones?
Have you read the thread you linked to, where several different definitions were mentioned, or are you just arguing to argue?