Chapter 13, Problem 83GP

To determine

To Find: The total number of air molecules.

Answer to Problem 83GP

  $1.37\times {10}^{44}$

Explanation of Solution

Given info:

Pressure, $P=1\text{ atm }=1.013\times {10}^5{\text{ N/m}}^{\text{2}}$

Boltzmann constant, $k=1.38\times {10}^{-23}\text{ J/K}$

Temperature, $T=0\text{°C }=273\text{ K}$

Volume, $V=\frac{4}{3}\times \pi \times \left(r_1{}^3-r_2{}^3\right)=5.11\times {10}^{18}{\text{ m}}^{\text{3}}$ .

Formula used:

Write the real gas equation.

  $\begin{array}{l}P\times V=N\times k\times T\\ N=\frac{P\times V}{k\times T}\end{array}$

Here, $P$ is the pressure of the air molecules,

  $V$ is the volume of the air molecules,

  $N$ is the number of molecules,

  $k$ is the Boltzmann constant,

  $T$ is the Temperature of the air.

Calculation:

Substitute the given values in the above equation,

  $\begin{array}{l}N=\frac{1.013\times {10}^5\times 5.11\times {10}^{18}}{1.38\times {10}^{-23}\times 273}\\ N=1.37\times {10}^{44}\text{ molecules}\end{array}$

Conclusion:

Thus, the total number of air molecules in the Earth's atmosphere is $1.37\times {10}^{44}$ .