91Ó°ÊÓ

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

${\left[c_v\right]}_e=\frac{{\mathrm{Ï€}}^{2}N{k}^{2}T}{2{\mathrm{Ï\mu }}_F}$

where,

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

$k=$Boltzmann constant

$T=$room Temperature

${\mathrm{Ï\mu }}_F\text{=the fermi energy.}$

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

role="math" localid="1647886513817" $\text{Puttingthevalue}6.022×{10}^{23}\text{for}N,8.617×{10}^{-5}\mathrm{eV}/\mathrm{K}\text{for}k,300\mathrm{K}\text{for}T\text{, and}7.05\mathrm{eV}\text{for}{\mathrm{Ï\mu }}_F$

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

$=9.389×{10}^{17}$

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

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

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

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

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

$T_D\text{=Debye temperature.}$

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

$\text{Duringthe higher temperature where}T≫T_D\text{, the specific heat is}$

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

$\text{Puttingthevalue}6.022×{10}^{23}\text{for}N\text{and}8.617×{10}^{-5}\mathrm{eV}/\mathrm{K}\text{for}k$

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

$=25\mathrm{J}/\mathrm{K}$

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

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

$=0.006$

$<1$

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

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