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Year 2 of the natural sciences course of the University of Cambridge

Chemistry A course of the University of Cambridge

Past exam of the chemistry A course of the University of Cambridge

2021

Paper 2

The equal supplied masses give equal <effective density of states>. At the specified <temperature>,
$$
N_c=N_v=2\left[\frac{2\pi(0.3m_e)k_{\rm B}(400\ {\rm K})}{h^2}\right]^{3/2}
\simeq6.35\times10^{24}\ {\rm m^{-3}}.
$$
Using the same <energy> units in the exponential,
$$
n_i=N_c e^{-E_g/(2k_{\rm B}T)}
\simeq(6.35\times10^{24})e^{-0.67/[2(8.61733\times10^{-5})(400)]}
\simeq3.82\times10^{20}\ {\rm m^{-3}}.
$$
For equal <electron> and hole mobilities, both carriers contribute to the intrinsic <electrical conductivity>:
$$
\boxed{\mu_e=\mu_h=\frac{\sigma}{2en_i}
\simeq\frac{48}{2(1.60218\times10^{-19})(3.82\times10^{20})}
\simeq0.392\ {\rm m^2\,V^{-1}\,s^{-1}}.}
$$
The factor two is essential. A dimensional check gives
$$
\frac{\mathrm{S\,m^{-1}}}{\mathrm{C\,m^{-3}}}
=\frac{\mathrm{A\,V^{-1}\,m^2}}{\mathrm C}
=\mathrm{m^2\,V^{-1}\,s^{-1}},
$$
using $\mathrm A=\mathrm{C\,s^{-1}}$.

Solution

Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A10 / a / ii / Solution

Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A10 / a / ii / Solution

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