In a certain region of space there are only 5 gaseous molecules per cm^3 on average. The temperature there is 3 K. The pressure of this gas is
`(k_(B)=1.38xx10^(-23Jmol^(-1)K^(-1)))`

Options

A. `20.7xx10^(-17)Nm^(-2)`
B. `20.4xx10^(-17)Nm^(-2)`
C. `10.7xx10^(-16)Nm^(-2)`
D. `10.7xx10^(-17)Nm^(-2)`

Explanation

To find the pressure of the gas in the given region of space, we can use the ideal gas equation in terms of the Boltzmann constant. The equation we will use is:

\[ PV = n k_B T \]

Where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of molecules,
- \( k_B \) is the Boltzmann constant, and
- \( T \) is the temperature.

### Step 1: Identify the given values
- Number of molecules per cm³, \( n = 5 \) molecules/cm³
- Temperature, \( T = 3 \) K
- Boltzmann constant, \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \)

### Step 2: Convert the number density to SI units
The number density \( n/V \) in SI units (m³) is given by converting the number of molecules per cm³ to molecules per m³. Since \( 1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3 \):

\[ n/V = 5 \, \text{molecules/cm}^3 = 5 \times 10^{6} \, \text{molecules/m}^3 \]

### Step 3: Substitute values into the pressure equation
We can rearrange the ideal gas equation to solve for pressure \( P \):

\[ P = \frac{n}{V} k_B T \]

Substituting the values we have:

\[ P = (5 \times 10^{6} \, \text{molecules/m}^3) \times (1.38 \times 10^{-23} \, \text{J/K}) \times (3 \, \text{K}) \]

### Step 4: Calculate the pressure
Calculating the above expression:

1. First calculate \( 5 \times 3 = 15 \).
2. Then multiply by \( 1.38 \times 10^{-23} \):

\[ P = 15 \times 1.38 \times 10^{-23} \times 10^{6} = 20.7 \times 10^{-17} \, \text{N/m}^2 \]

### Step 5: Final result
Thus, the pressure of the gas is:

\[ P = 2.07 \times 10^{-16} \, \text{N/m}^2 \]