The mass aof `350cm^(3)` of a diatomic gas at 273 K at 2 atmospheres pressure is one gram. Calculate the mass of one atom of the gas.

Explanation

To calculate the mass of one atom of a diatomic gas given the conditions, we can follow these steps:

### Step 1: Use the Ideal Gas Law
The Ideal Gas Law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles
- \( R \) = universal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature (in Kelvin)

### Step 2: Convert Volume to Liters
The volume given is \( 350 \, \text{cm}^3 \). To convert this to liters:
\[ V = \frac{350 \, \text{cm}^3}{1000} = 0.350 \, \text{L} \]

### Step 3: Substitute Values into the Ideal Gas Law
Now, substituting the known values into the Ideal Gas Law:
- \( P = 2 \, \text{atm} \)
- \( V = 0.350 \, \text{L} \)
- \( R = 0.0821 \, \text{L·atm/(K·mol)} \)
- \( T = 273 \, \text{K} \)

The equation becomes:
\[ 2 \times 0.350 = n \times 0.0821 \times 273 \]

### Step 4: Solve for \( n \) (Number of Moles)
Rearranging the equation to solve for \( n \):
\[ n = \frac{2 \times 0.350}{0.0821 \times 273} \]

Calculating this gives:
\[ n \approx 0.0315 \, \text{moles} \]

### Step 5: Calculate the Number of Atoms
Since the gas is diatomic, each molecule contains 2 atoms. Therefore, the number of molecules in \( n \) moles is given by:
\[ \text{Number of molecules} = n \times N_A \]
Where \( N_A = 6.022 \times 10^{23} \, \text{molecules/mol} \).

Thus, the number of molecules is:
\[ \text{Number of molecules} = 0.0315 \times 6.022 \times 10^{23} \]

### Step 6: Calculate the Mass of One Atom
The total mass of the gas is given as 1 gram. To find the mass of one atom, we first find the total number of atoms:
\[ \text{Total number of atoms} = 2 \times \text{Number of molecules} \]

Now, the mass of one atom is calculated as:
\[ \text{Mass of one atom} = \frac{\text{Total mass}}{\text{Total number of atoms}} \]

Substituting the values:
\[ \text{Mass of one atom} = \frac{1 \, \text{g}}{2 \times (0.0315 \times 6.022 \times 10^{23})} \]

### Step 7: Final Calculation
Calculating this gives:
\[ \text{Mass of one atom} \approx 5.3 \times 10^{-23} \, \text{g} \]

### Final Answer:
The mass of one atom of the diatomic gas is approximately \( 5.3 \times 10^{-23} \, \text{g} \).

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