# Size of a cube for a molecule of ideal gas

• Karol
Ideal Gas Law, we can determine the side length of a cube containing ideal gas molecules at 0°C and 1 atm pressure. In summary, the side length is approximately 3.1 angstroms, with typical molecules being about 3-4 angstroms in size.

## Homework Statement

The temperature of an ideal gas is 00C and the pressure is 1[atm]. imagine every molecule is enclosed in a cube, what's it's side length?

PV=nRT

## The Attempt at a Solution

I assume volume of i liter:
$$1[atm]\cdot 1[liter]=n\cdot 0.08208\cdot 273\rightarrow n=0.0446[mole]$$
Molecules per 1 liter:
$$0.0446\cdot 6.023\cdot 10^{23}=2.687\cdot 10^{22}$$
How many molecules are on one side?
$$\sqrt[3]{2.687\cdot 10^{22}}=29951774$$
The length of a side:
$$\frac{10[cm]}{2995177}=3.34\cdot 10^{-7}[cm]$$
The answer in the book: 3E-7[cm]

No specification of significant figures anywhere in the problem statement? Haven't checked your arithmetic in detail, but the set-up and execution looks great.

I always like to draw a picture of what the molecule in the box looks like when I teach this, to give students a sense of scale.

The box is about 30 Angstrom on a side. How big is a typical molecule? How big is the typical box for a liquid?

A bi atomic molecule is about 1[A] am i right? then the side is bigger 30 times more, but in the book it's written that's it's only 10 times larger.
The volume of one mole of water is 18[cm3]. Molecular weight 18:
$$\sqrt[3]{6.023\times 10^{23}}=84450901,\ \sqrt[3]{18}=2.62[cm]$$
$$\frac{2.62}{84450901}=31\times10^{-9}[cm]=31\times 10^{-11}[m]=3.1\times 10^{-10}[m]=3.1[Angstram]$$
Water molecule's size is about 1.5[A]

Water molecule is a bit bigger (diameter 2.75 A here); O-H centers are .94 Angstrom apart.

And here are a few other molecules. 3 to 4 A appears to be a good estimate

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Thanks