In a HCl molecule, we may treat Cl to be of infinite mass and H alone oscillating. If the oscillations of HCI molecule shows frequency `9 xx 10^(13)s^(-1)` , deduce the force constant. Given Avogadro's number `6 xx 10^(26)` per kg mole.

Explanation

To solve the problem, we need to calculate the force constant (k) for the oscillating hydrogen atom in an HCl molecule, given the frequency of oscillation and Avogadro's number. Here’s a step-by-step solution:

### Step 1: Understand the relationship between frequency, angular frequency, and force constant
The angular frequency (ω) is related to the frequency (f) by the equation:
\[ \omega = 2\pi f \]
The force constant (k) is related to the angular frequency and the mass (m) of the oscillating particle by:
\[ \omega = \sqrt{\frac{k}{m}} \]

### Step 2: Rearranging the equations
By substituting the expression for ω into the equation for k, we get:
\[ \omega^2 = \frac{k}{m} \implies k = m\omega^2 \]
Substituting ω from the first equation:
\[ k = m(2\pi f)^2 \]

### Step 3: Calculate the mass of the hydrogen atom
Given Avogadro's number (N_A = \(6 \times 10^{26}\) per kg mole), we can find the mass of one hydrogen atom (m_H) in kilograms:
\[ m_H = \frac{1 \text{ g/mol}}{N_A} = \frac{1 \times 10^{-3} \text{ kg}}{6 \times 10^{26}} \approx 1.66 \times 10^{-27} \text{ kg} \]

### Step 4: Substitute values into the equation for k
Now, we substitute the values into the equation for k:
\[ f = 9 \times 10^{13} \text{ s}^{-1} \]
\[ k = m_H (2\pi f)^2 \]
Calculating \(2\pi f\):
\[ 2\pi f = 2 \times \frac{22}{7} \times (9 \times 10^{13}) \approx 5.654 \times 10^{14} \text{ s}^{-1} \]
Now squaring it:
\[ (2\pi f)^2 \approx (5.654 \times 10^{14})^2 \approx 3.20 \times 10^{29} \text{ s}^{-2} \]

### Step 5: Calculate the force constant k
Now, substituting the values into the equation for k:
\[ k = (1.66 \times 10^{-27} \text{ kg}) \times (3.20 \times 10^{29} \text{ s}^{-2}) \approx 5.31 \text{ N/m} \]

### Final Result
The force constant \(k\) is approximately:
\[ k \approx 5.31 \text{ N/m} \]