Chapter 7: Q. 7.58 (page 313)

The speed of sound in copper is $3560 m / s$ . Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.

Short Answer

Expert verified

Hence, the Debye Temperature from experimental Data (graph) $T_{D} = 359.9176 K$ .

Step by step solution

Step 1. Given information

We have, $the speed of sound in copper is 3560 m / s$

$The Debye temperature is T_{D} = \frac{h C_{s}}{2 k_{B}} \cdot {(\frac{6 N}{π V})}^{\frac{1}{3}}$ .

Step 2. Putting the value of h , V , N ,CS ,kB.

$V = 7.11 {cm}^{3} / mol$

$C_{s} = 3560 m / s = 3560 \times 100 cm / s$

$N = 6.022 \times 10^{23} atoms / mol$

$k_{B} = 1.381 \times 10^{- 23} J / K$

$h = 6.626 \times 10^{- 34} J \cdot s$

substituting all the above value in Debye Temperature we get

$T_{D} = \frac{6.626 \times 10^{- 34} J \cdot s \times 3560 \times 100 cm / s}{(2 \times 1.381 \times 10^{- 23} J / K)} {[\frac{6 \times 6.022 \times 10^{23} / mol}{π \times 7.11 {cm}^{3} / mol}]}^{\frac{1}{3}}$

$∴ T_{D} = 465.3381 K$

Step 3. Calculating the experimental Debye Temperature.

Take any two points on experimental data calculate the slope [Approximately] Approximately take two points

Approximately take two points

$A (0, 0.75) B (18, 1.5)$

$\frac{Δ (\frac{C_{V}}{T})}{Δ (T^{2})} = \frac{1.5 - 0.75}{18 - 0}$

$= 0.0417$

Step 4. Finding the slope of the graph.

$Slope: \frac{Δ (\frac{C_{V}}{T})}{Δ (T^{2})} = \frac{12 π^{4} N k_{B}}{5 T_{D}^{3}}$

$\frac{12 π^{4} N k_{B}}{5 T_{D}^{3}} = 0.0417 mJ / K^{4}$

then , the Debye Temperature

$T_{D} = {[\frac{12 π^{4} N k_{B}}{5 \times 0.0417 \times 10^{- 3} J / K^{4}}]}^{\frac{1}{3}}$

$= {(\frac{12 π^{4} \times 6.022 \times 10^{23} \times 1.381 \times 10^{- 23}}{5 \times 0.0417 \times 10^{- 3}})}^{\frac{1}{3}}$

$= 359.9176 K$

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The speed of sound in copper is $3560 m / s$ . Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Putting the value of h , V , N ,CS ,kB.

Step 3. Calculating the experimental Debye Temperature.

Step 4. Finding the slope of the graph.

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The speed of sound in copper is 3560m/s. Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Putting the value of h , V , N ,CS ,kB.

Step 3. Calculating the experimental Debye Temperature.

Step 4. Finding the slope of the graph.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Energy Physics

Electricity

Space Physics

Solid State Physics

Physics of Motion

Study anywhere. Anytime. Across all devices.

The speed of sound in copper is $3560 m / s$ . Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.