A monoatomic gas is kept in a 1 litre container at pressure 1 atm. If average energy per molecule is `2 x 10^-9 J`. Find number of molecules in the container.

Options

A. `0.75 x 10^-6`
B. `0.75 x 10^9`
C. `0.5 x 10^-11`
D. `0.75 x 10^11`

Answer

To find the number of molecules in a monoatomic gas contained in a 1-liter container at a pressure of 1 atm, with an average energy per molecule of \(2 \times 10^{-9} \, J\), we can follow these steps:

### Step 1: Understand the relationship between total energy and average energy per molecule
The total energy \(E\) of the gas can be expressed in terms of the number of molecules \(N\) and the average energy per molecule \(E_{avg}\):
\[ E = N \cdot E_{avg} \]
Where \(E_{avg} = 2 \times 10^{-9} \, J\).

### Step 2: Use the formula for the total energy of a monoatomic gas
For a monoatomic ideal gas, the total energy can also be expressed as:
\[ E = \frac{3}{2} nRT \]
Where \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. However, we can also relate this to pressure and volume using the ideal gas law:
\[ PV = nRT \]
Thus, we can express the total energy as:
\[ E = \frac{3}{2} PV \]

### Step 3: Convert units
We need to convert the pressure and volume to SI units:
- Pressure \(P = 1 \, atm = 1.013 \times 10^5 \, Pa\)
- Volume \(V = 1 \, liter = 1 \times 10^{-3} \, m^3\)

### Step 4: Calculate the total energy using pressure and volume
Now we can calculate the total energy:
\[ E = \frac{3}{2} PV = \frac{3}{2} \times (1.013 \times 10^5 \, Pa) \times (1 \times 10^{-3} \, m^3) \]
Calculating this gives:
\[ E = \frac{3}{2} \times 1.013 \times 10^5 \times 10^{-3} = \frac{3}{2} \times 101.3 \, J = 151.95 \, J \]

### Step 5: Relate total energy to number of molecules
From the relationship established in Step 1, we can set:
\[ 151.95 = N \cdot (2 \times 10^{-9}) \]
Now, we can solve for \(N\):
\[ N = \frac{151.95}{2 \times 10^{-9}} = 7.5975 \times 10^{10} \]

### Step 6: Finalize the answer
Rounding this value gives:
\[ N \approx 0.76 \times 10^{11} \text{ molecules} \]

Thus, the number of molecules in the container is approximately \(0.76 \times 10^{11}\).