The vander waals constatn 'b' of a gas is `4 pi xx 10^(-4) L//mol`. The radius of gas atom can be expressed in secientific notation as `z = 10^(-9)cm`. Calculate the value fo `z`.(Given `N_(A) = 6 xx 10^(23))`

Answer

To solve the problem, we will use the relationship between the Van der Waals constant 'b', Avogadro's number, and the volume of a single atom. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship for 'b'
The Van der Waals constant 'b' is related to the volume occupied by one mole of gas and can be expressed as:
\[ b = N_A \cdot V_{\text{atom}} \]
where \( V_{\text{atom}} \) is the volume of one atom and \( N_A \) is Avogadro's number.

### Step 2: Express the volume of one atom
The volume of a single atom can be approximated as the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the atom.

### Step 3: Substitute into the equation
Substituting the expression for \( V_{\text{atom}} \) into the equation for 'b':
\[ b = N_A \cdot \left(\frac{4}{3} \pi r^3\right) \]

### Step 4: Rearrange the equation
We can rearrange the equation to solve for \( r^3 \):
\[ r^3 = \frac{3b}{4 \pi N_A} \]

### Step 5: Substitute known values
Given:
- \( b = 4 \pi \times 10^{-4} \, \text{L/mol} \)
- \( N_A = 6 \times 10^{23} \, \text{mol}^{-1} \)

Substituting these values into the equation:
\[ r^3 = \frac{3 \times (4 \pi \times 10^{-4})}{4 \pi \times (6 \times 10^{23})} \]

### Step 6: Simplify the equation
The \( 4 \pi \) terms cancel out:
\[ r^3 = \frac{3 \times 10^{-4}}{6 \times 10^{23}} \]
\[ r^3 = \frac{1}{2} \times 10^{-4} \times 10^{-23} \]
\[ r^3 = \frac{1}{2} \times 10^{-27} \]

### Step 7: Calculate \( r \)
Taking the cube root:
\[ r = \left(\frac{1}{2} \times 10^{-27}\right)^{1/3} \]
\[ r \approx 1.077 \times 10^{-9} \, \text{cm} \]

### Step 8: Express in scientific notation
Thus, we can express the radius \( r \) as:
\[ z = 1.077 \times 10^{-9} \, \text{cm} \]

### Final Answer
The value of \( z \) is:
\[ z = 1.077 \times 10^{-9} \, \text{cm} \]

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