How many air molecules are in a room at temperature `23.8^@ C` and standard pressure if the dimensions of the room are `3.66 m xx 3.66 m xx 2.43 m`?

Options

A. 1330
B. 16600
C. `3.03 xx 10^(24)`
D. `8.05 xx 10^(26)`

Answer

To find the number of air molecules in a room at a temperature of 23.8°C and standard pressure, we can follow these steps:

### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Kelvin (K) can be calculated using the formula:
\[ T(K) = T(°C) + 273.15 \]
Given \( T = 23.8°C \):
\[ T = 23.8 + 273.15 = 296.95 K \]

### Step 2: Calculate the volume of the room
The volume \( V \) of a cuboidal room can be calculated using the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given dimensions:
- Length = 3.66 m
- Breadth = 3.66 m
- Height = 2.43 m

Calculating the volume:
\[ V = 3.66 \times 3.66 \times 2.43 \]
\[ V = 34.24 \, m^3 \]

### Step 3: Use the Ideal Gas Law to find the number of moles
The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \( P \) = pressure (standard pressure = \( 1.01 \times 10^5 \, Pa \))
- \( V \) = volume (calculated above)
- \( n \) = number of moles
- \( R \) = universal gas constant (\( 8.314 \, J/(mol \cdot K) \))
- \( T \) = temperature in Kelvin (calculated above)

Rearranging the equation to solve for \( n \):
\[ n = \frac{PV}{RT} \]

Substituting the values:
\[ n = \frac{(1.01 \times 10^5 \, Pa)(34.24 \, m^3)}{(8.314 \, J/(mol \cdot K))(296.95 \, K)} \]

Calculating \( n \):
\[ n = \frac{3464240}{2468.15} \]
\[ n \approx 1402.11 \, moles \]

### Step 4: Calculate the number of molecules
Using Avogadro's number (\( 6.022 \times 10^{23} \, molecules/mol \)), we can find the total number of molecules:
\[ \text{Number of molecules} = n \times N_A \]
Where \( N_A \) is Avogadro's number.

Substituting the values:
\[ \text{Number of molecules} = 1402.11 \times 6.022 \times 10^{23} \]
\[ \text{Number of molecules} \approx 8.44 \times 10^{26} \]

### Final Answer
The total number of air molecules in the room is approximately \( 8.44 \times 10^{26} \) molecules.

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