Question

The Debye temperatures ($\theta_D$) of diamond and copper are 2230 K and 343 K, respectivelyThe maximum value of the frequency of the atomic oscillations of diamond is Y times the maximum value of the frequency of the atomic oscillations of copperThe value of Y is _ _ _. (rounded off to two decimal places)

Hint:

Debye temperature is directly proportional to maximum vibrational frequencyHigher $\theta_D$ means stiffer lattice and higher frequency

Answer 1

Correct Answer: 6.5

Step 1: Relation between Debye temperature and frequency.
\[ \theta_D = \frac{h\nu_{\max}}{k_B} \]
Thus,
\[ \nu_{\max} \propto \theta_D \]

Step 2: Write ratio of maximum frequencies.
\[ Y = \frac{\nu_{\max}(\text{diamond})}{\nu_{\max}(\text{copper})} = \frac{\theta_D(\text{diamond})}{\theta_D(\text{copper})} \]

Step 3: Substitute given values.
\[ Y = \frac{2230}{343} \]

Step 4: Calculate ratio.
\[ Y \approx 6.50 \]

Step 5: Conclusion.
\[ \boxed{6.50} \]