Chapter 1.2, Problem 99E

Number of Molecules A sealed room in a hospital, measuring 5 m wide, 10 m long, and 3 m high, is filled with pure oxygen. One cubic meter contains 1000 L, and 22.4 L of any gas contains 6.02 × 10²³ molecules (Avogadro’s number). How many molecules of oxygen are there in the room?

To determine

The number of molecules of oxygen in the room.

Answer to Problem 99E

The number of molecules of oxygen in the room is $\underset{\_}{4.03\times {10}^{27}\text{molecules}}$.

Explanation of Solution

Formula used:

$\text{Volume}\text{of}\text{a}\text{sealed}\text{room}=\text{Length}\times \text{Width}\times \text{Depth}$

$\text{Number}\text{of}\text{molecules}\text{in}\text{a}\text{sealed}\text{room}=\left(\begin{array}{l}\text{Volume}\text{of}\text{a}\\ \text{sealed}\text{room}\left(\text{L}\right)\end{array}\right)\times \left(\begin{array}{l}\text{Number}\text{of}\\ \text{molecules}\text{in}\text{1}\text{L}\end{array}\right)$

Calculation:

Given that, the length of a sealed room in a hospital is $10\text{m}$, the width of a sealed room in a hospital is $5\text{m}$ and the depth of a sealed room in a hospital is $3\text{m}$.

It is known that, one cubic meter contains 1000 L and 22.4 L of any gas contains $6.02\times {10}^{23}\text{molecules}$.

Substitute $10\text{m}$ for Length, $5\text{m}$ for Width and $3\text{m}$ for Depth $3.7\times {10}^3\text{m}$ in the formula to calculate volume of a sealed room.

$\begin{array}{c}\text{Volume}\text{of}\text{a}\text{sealed}\text{room}=10\text{m}\times 5\text{m}\times 3\text{m}\\ =150{\text{m}}^3\end{array}$

Use the unit conversion of ${\text{m}}^3$ into L to convert volume of a sealed room in ${\text{m}}^3$ into L.

$\begin{array}{c}\text{Volume}\text{of}\text{a}\text{sealed}\text{room}=150{\text{m}}^3\\ =150\times 1{\text{m}}^3\\ =150\times 1000\text{L}\\ \text{=150000}\text{L}\end{array}$

Calculate the number of molecules in 22.4 L as follows.

$\begin{array}{l}22.4\text{L}=6.02\times {10}^{23}\text{molecules}\\ \text{1L}=\frac{6.02\times {10}^{23}\text{molecules}}{22.4\text{L}}\\ \text{1L}=0.26875\times {10}^{23}\text{molecules}\end{array}$

Therefore, the number of molecules in 1 L is $0.26875\times {10}^{23}\text{molecules}$.

Substitute $150000\text{L}$ for volume of a sealed room and $0.26875\times {10}^{23}\text{molecules}$ for number of molecules in 1 L.

$\begin{array}{c}\text{Number of molecules in a sealed room}=\left(150000\text{L}\right)\times \left(0.26875\times {10}^{23}\text{molecules}\right)\\ =40312.5\times {10}^{23}\text{molecules}\end{array}$

Now, move decimal 4 places to the left and add 4 to the power of 10 to write the above quantity in proper scientific notation as follows.

$\begin{array}{c}\text{Number}\text{of}\text{molecules}\text{in}\text{a}\text{sealed}\text{room}=40312.5\times {10}^{23}\text{molecules}\\ =4.03125\times {10}^{27}\text{molecules}\end{array}$

Round off the above quantity to two decimal places.

$\begin{array}{c}\text{Number}\text{of}\text{molecules}\text{in}\text{a}\text{sealed}\text{room}=4.03125\times {10}^{27}\text{molecules}\\ =4.03\times {10}^{27}\text{molecules}\end{array}$

Therefore, the number of molecules of oxygen in the room is $\underset{\_}{4.03\times {10}^{27}\text{molecules}}$.