Example 13.7 from College Physics, 13.3 The Ideal Gas Law

Calculating the Number of Molecules in a Cubic Meter of Gas

How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large $N$ typically is.

Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be $0 º C$ and atmospheric pressure.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Because pressure, volume, and temperature are all specified, we can use the ideal gas law $PV = NkT$ , to find $N$ .
Solution
1. Identify the knowns.

$\begin{array}{l} T & = & 0 º C = 273 K \\ P & = & 1 . 01 \times 10^{5} Pa \\ V & = & 1.00 m^{3} \\ k & = & 1 . 38 \times 10^{- 23} J/K \end{array}$
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2. Identify the unknown: number of molecules, $N$ .

3. Rearrange the ideal gas law to solve for $N$ .

$\begin{matrix} PV = NkT \\ N = \frac{PV}{kT} \end{matrix}$
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4. Substitute the known values into the equation and solve for $N$ .

$N = \frac{PV}{kT} = \frac{(1 . 01 \times 10^{5} Pa) (1 . {00 m}^{3})}{(1 . 38 \times 10^{- 23} J/K) (273 K)} = 2 . 68 \times 10^{25} molecules$
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Discussion
This number is undeniably large, considering that a gas is mostly empty space. $N$ is huge, even in small volumes. For example, $1 {cm}^{3}$ of a gas at STP has $2.68 \times 10^{19}$ molecules in it. Once again, note that $N$ is the same for all types or mixtures of gases.

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Strategy

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