There are `4 xx 10^(24)` gas molecules in a vessle at `50 K` temperature. The pressure of the gas in the vessel is `0.03 atm`. Calculate the volume of the vessel.

Answer

To solve the problem of calculating the volume of the vessel containing gas molecules, we will use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure (in atm)
- \( V \) = Volume (in m³)
- \( n \) = Number of moles of gas
- \( R \) = Ideal gas constant (in appropriate units)
- \( T \) = Temperature (in Kelvin)

### Step-by-Step Solution:

**Step 1: Calculate the number of moles of gas.**
The number of moles \( n \) can be calculated using the formula:

\[ n = \frac{N}{N_A} \]

Where:
- \( N \) = Number of molecules = \( 4 \times 10^{24} \)
- \( N_A \) = Avogadro's number = \( 6.022 \times 10^{23} \, \text{molecules/mol} \)

Calculating \( n \):

\[ n = \frac{4 \times 10^{24}}{6.022 \times 10^{23}} \approx 6.64 \, \text{moles} \]

**Step 2: Convert pressure from atm to pascals (N/m²).**
The pressure given is \( 0.03 \, \text{atm} \). To convert this to pascals, we use the conversion factor \( 1 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2 \):

\[ P = 0.03 \, \text{atm} \times 1.013 \times 10^5 \, \text{N/m}^2/\text{atm} \approx 3040 \, \text{N/m}^2 \]

**Step 3: Use the Ideal Gas Law to find the volume.**
Rearranging the Ideal Gas Law to solve for volume \( V \):

\[ V = \frac{nRT}{P} \]

Where:
- \( R = 8.31 \, \text{J/(mol K)} \)
- \( T = 50 \, \text{K} \)

Substituting the values into the equation:

\[ V = \frac{(6.64 \, \text{mol}) \times (8.31 \, \text{J/(mol K)}) \times (50 \, \text{K})}{3040 \, \text{N/m}^2} \]

Calculating \( V \):

\[ V = \frac{(6.64 \times 8.31 \times 50)}{3040} \approx \frac{2764.2}{3040} \approx 0.91 \, \text{m}^3 \]

### Final Answer:
The volume of the vessel is approximately \( 0.919 \, \text{m}^3 \).

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