A gas at 300 K has pressure `4 xx 10^(-10) N//m^(2)`. IF `k = 1.38 xx 10^(-23) J//K`, the number of `"molecule"// cm^(3)` is of the order of

Options

A. `10^(5)`
B. 10
C. `10^(7)`
D. `10^(11)`

Explanation

To solve the problem of finding the number of molecules per cubic centimeter for the given gas, we can use the ideal gas equation in conjunction with the Boltzmann constant. Here’s a step-by-step breakdown of the solution:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Temperature (T) = 300 K
- Pressure (P) = \(4 \times 10^{-10} \, \text{N/m}^2\)
- Boltzmann constant (k) = \(1.38 \times 10^{-23} \, \text{J/K}\)

2. **Use the Ideal Gas Law:**
The ideal gas law can be expressed in terms of the number of molecules:
\[ PV = N k T \]
where \(N\) is the number of molecules, \(P\) is the pressure, \(V\) is the volume, \(k\) is the Boltzmann constant, and \(T\) is the temperature.

3. **Rearranging the Equation:**
To find the number of molecules per unit volume (molecules per cubic meter), we can rearrange the equation:
\[ N = \frac{PV}{kT} \]
Therefore, the number of molecules per unit volume (molecules/m³) is:
\[ n = \frac{N}{V} = \frac{P}{kT} \]

4. **Substituting the Values:**
Substitute the values into the equation:
\[ n = \frac{4 \times 10^{-10}}{(1.38 \times 10^{-23})(300)} \]

5. **Calculating the Denominator:**
First, calculate \(kT\):
\[ kT = (1.38 \times 10^{-23})(300) = 4.14 \times 10^{-21} \, \text{J} \]

6. **Calculating the Number of Molecules per Cubic Meter:**
Now substitute \(kT\) back into the equation for \(n\):
\[ n = \frac{4 \times 10^{-10}}{4.14 \times 10^{-21}} \approx 9.66 \times 10^{10} \, \text{molecules/m}^3 \]

7. **Convert to Molecules per Cubic Centimeter:**
Since \(1 \, \text{m}^3 = 10^6 \, \text{cm}^3\), we convert:
\[ n \approx \frac{9.66 \times 10^{10}}{10^6} \approx 9.66 \times 10^4 \, \text{molecules/cm}^3 \]

8. **Final Answer:**
The number of molecules per cubic centimeter is of the order of \(10^5\).