Q. 7.25 - An Introduction to Thermal Physics

Question

Use the results of this section to estimate the contribution of conduction electrons to the heat capacity of one mole of copper at room temperature. How does this contribution compare to that of lattice vibrations, assuming that these are not frozen out? (The electronic contribution has been measured at low temperatures, and turns out to be about$40\%$ more than predicted by the free electron model used here.)

Step-by-Step Solution

Answer

The electrons contribute less than  $1\%$ of the total heat capacity at room temperature. 

Step 1. Given information

The contribution of conduction electrons to the heat capacity of one mole of copper at room temperature is given as

${\left[c_v\right]}_e=\frac{{\pi }^{2}N{k}^{2}T}{2{ϵ}_F}$

where,

$N =$the number of atoms and is equal to Avogadro number of atoms per one mole .

 $k =$ Boltzmann constant

$T =$room Temperature

${ϵ}_F\text{ = the fermi energy. }$

Step 2. Calculating the value of c v e

${ϵ}_F=7.05\mathrm{eV}$ {The fermi energy of copper }

$\text{Putting the value }6.022\times {10}^{23}\text{ for }N,8.617\times {10}^{-5}\mathrm{eV}/\mathrm{K}\text{ for }k,300 \mathrm{K}\text{ for }T\text{, and }7.05\mathrm{eV}\text{ for }{ϵ}_F$

${\left[C_V\right]}_e=\frac{{\pi }^2\left(6.022\times {10}^{23}\right){\left(8.617\times {10}^{-5}\mathrm{eV}/\mathrm{K}\right)}^2(300 \mathrm{K})}{2(7.05\mathrm{eV})}$

          $=9.389\times {10}^{17}$

          $=\left(9.389\times {10}^{17}\mathrm{eV}/\mathrm{K}\right)\left(1.6\times {10}^{-19} \mathrm{J}/\mathrm{K}\right)$

          $=0.15 \mathrm{J}/\mathrm{K}$

So, the contribution of conduction electrons to the heat capacity of one mole of copper at room temperature is $0.15 \mathrm{J}/\mathrm{K}$.

Step 3. Calculating the value of C V l which is the specific heat due to lattice vibration.

 According to Debye theory of lattice vibrations, specific heat is given as 

${\left[C_V\right]}_l=\frac{12{\pi }^4}{5}{\left(\frac{T}{T_D}\right)}^{3}Nk$

$T_D\text{ = Debye temperature. }$

$\text{ The above formula is applicable when }T\ll <T_D\text{. }$

$\text{ During the higher temperature where }T\gg T_D\text{, the specific heat is }$

${\left[C_V\right]}_I=3Nk$

$\text{Putting the value }6.022\times {10}^{23}\text{ for }N\text{ and }8.617\times {10}^{-5}\mathrm{eV}/\mathrm{K}\text{ for }k$

${\left[C_V\right]}_{/}=3\left(6.022\times {10}^{23}\right)\left(1.381\times {10}^{-23} \mathrm{J}/\mathrm{K}\right)$

          $=25 \mathrm{J}/\mathrm{K}$

Step 4. Calculating the ratio of the contribution of electrons to the heat capacity of the lattice vibrations at room temperature.

So, the contribution of electrons is very small as compared to the heat capacity of the lattice vibrations at room temperature.

$\frac{{\left[C_V\right]}_e}{{\left[C_V\right]}_1}=\frac{0.15 \mathrm{J}/\mathrm{K}}{25 \mathrm{J}/\mathrm{K}}$

            $=0.006$

            $<1$

${\left[C_V\right]}_e<{\left[C_V\right]}_I$

$\text{So, the electrons contribute less than }1\%\text{ of the total heat capacity at room temperature. }$