Question

What is the density of water vapor at 25 C?

Answer 1

Well, by assuming ideality, we can use the ideal gas law.

`PM = DRT`,

where:

• `P` is pressure in `"bar"`.
• `M` is molar mass in `"g/mol"`.
• `R = 0.083145` is the universal gas constant in `"L"cdot"bar/mol"cdot"K"`.
• `T` is temperature in `"K"`.
• Hence the density `D` shall be in `"g/L"`.

And we obtain:

`color(blue)(D) = (PM)/(RT)`

`= (("1 bar")("18.015 g/mol"))/(("0.083145 L"cdot"bar/mol"cdot"K")("298.15 K"))`

`=` `color(blue)ul"0.7267 g/L"`

---

If not assuming ideality, it can be estimated by the van der Waals equation of state:

`P = (RT)/(barV - b) - a/barV^2`,

where:

• `barV = V/n` is the molar volume.
• `a` accounts for intermolecular forces, and shall be in units of `"bar"cdot"L"^2//"mol"^2`.
• `b` is the excluded volume, and shall be in units of `"L/mol"`.

The van der Waals constants of water are `a = "5.536 bar"cdot"L"^2//"mol"^2` and `b = "0.03049 L/mol"`. To solve for the mass density, we would have to rearrange this into the cubic molar volume form.

Get common denominators:

`P = (RTbarV^2 - a(barV - b))/((barV - b)(barV^2))`

Multiply through...

`P(barV - b)(barV^2) = RTbarV^2 - a(barV - b)`

Distribute...

`PbarV^3 - bP barV^2 = RTbarV^2 - abarV + ab`

Rearrange to get:

`ul(PbarV^3 - (bP + RT) barV^2 + abarV - ab = 0)`

This is in general difficult to solve, but an iterative method exists called the Newton-Raphson method:

`x_(n+1) = x_n - (f(x_n))/(f'(x_n))`

In this case, `x_n = barV_n`, and `f(barV_n)` is the cubic equation shown above. The form of the ratio is:

`(f(x_n))/(f'(x_n)) = (PbarV^3 - (bP + RT) barV^2 + abarV - ab)/(3PbarV^2 - 2(bP + RT) barV + a)`

So, to solve this, let:

• `barV = X = "take a guess number"`
• `P = 1`
• `bP + RT = 24.82017175`
• `a = 5.536`
• `ab = 0.16879264`

with the appropriate units. Thus, put the following into your TI calculator to solve for `barV` at `25^@ "C"`, or `"298.15 K"`:

`(X - (X^3 - 24.82017175X^2 + 5.536X - 0.16879264)/(3X^2 - 49.6403435X + 5.536)) -> X`

Then, press enter until your answer converges. There are three "answers" you would get:

• One molar volume for the liquid phase
• One molar volume for the gas phase
• One molar volume in between them that makes no physical sense.

I got it to converge by taking some guesses...

• Starting at `x = 0.05`, I get a molar volume of `"24.5954 L/mol"`, or a mass density of `color(blue)ul"0.7325 g/L"`.
• Starting at `x = 5`, I get `"0.188372 L/mol"`.
• Starting at `x = 20`, I get `"0.0364321 L/mol"`.

The first answer is for the gas, and that's all we're looking for here. (I got the density as

`D = M//barV`.)