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^(3)`
B. `10^(5)`
C. `10^(6)`
D. `10^(9)`

Explanation

To find the number of molecules per cubic centimeter in a gas at a given temperature and pressure, we can use the relation derived from the ideal gas law. Here’s a step-by-step solution:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by:
\[ PV = nRT \]
where:
- \( P \) = pressure,
- \( V \) = volume,
- \( n \) = number of moles,
- \( R \) = universal gas constant,
- \( T \) = temperature in Kelvin.

### Step 2: Relate Moles to Molecules
We know that:
\[ n = \frac{N}{N_A} \]
where \( N \) is the number of molecules and \( N_A \) is Avogadro's number.

### Step 3: Use Boltzmann's Constant
We can express the ideal gas law in terms of the number of molecules:
\[ PV = NkT \]
where \( k \) is Boltzmann's constant. Rearranging gives:
\[ N = \frac{PV}{kT} \]

### Step 4: Convert Units
Since we need the number of molecules per cubic centimeter, we need to convert the pressure from \( N/m^2 \) to \( N/cm^2 \):
\[ 1 \, m^2 = 10^4 \, cm^2 \]
Thus, the pressure in \( N/cm^2 \) is:
\[ P = 4 \times 10^{-10} \, N/m^2 = 4 \times 10^{-10} \, N/m^2 \times \frac{1}{10^4} = 4 \times 10^{-14} \, N/cm^2 \]

### Step 5: Substitute Values
Now, substituting the values into the equation:
\[ N = \frac{P}{kT} \]
where:
- \( P = 4 \times 10^{-14} \, N/cm^2 \),
- \( k = 1.38 \times 10^{-23} \, J/K \),
- \( T = 300 \, K \).

### Step 6: Calculate \( N \)
Now we can calculate \( N \):
\[ N = \frac{4 \times 10^{-14}}{(1.38 \times 10^{-23})(300)} \]
Calculating the denominator:
\[ 1.38 \times 10^{-23} \times 300 = 4.14 \times 10^{-21} \]
Now substituting back:
\[ N = \frac{4 \times 10^{-14}}{4.14 \times 10^{-21}} \approx 9.66 \times 10^{6} \, molecules/cm^3 \]

### Step 7: Order of Magnitude
The order of magnitude of \( N \) is approximately \( 10^7 \) molecules/cm³.

### Final Answer
Thus, the number of molecules per cubic centimeter is of the order of \( 10^7 \).