Question

In a gas at standard conditions, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about$\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{9}}$people)?

Short answer

$6.39\times {10}^{-6}m$ is the length of the side of the cube.

Step-by-step solution

Calculation of volume

Given data:

$T=273K,P=1.01\times {10}^{5}Pa,N=7\times {10}^9$

Putting the equation;

$$\begin{gathered} V=\frac{N\times K\times T}{P} \\ =\frac{7\times {10}^9\times 1.381\times {10}^{-23}\times 273}{1.01\times {10}^5} \\ V=2.61\times {10}^{-16}{m}^3 \end{gathered}$$

Calculation of length

Now the length is

$$\begin{gathered} L^3=V \\ L=\sqrt[3]{V} \\ =\sqrt[3]{2.61\times {10}^{-16}} \\ =6.39\times {10}^{-6}m \end{gathered}$$

$6.39\times {10}^{-6}m$is the length of the side of the cube.