Question

The number of atoms in volume of one cubic
meter of hydrogen gas is 9.8 1026. The radius
of the hydrogen atom is 0.53 Å. Calculate the
polarizability and relative permittivity.
(€,= 8-85 x 10-12 F/m)​

Answer 1

Answer:

The polarizability is equal to 16.548×10⁻⁴² Fm².

The relative permittivity is equal to 1.0018316.

Explanation:

The number of atoms of hydrogen gas, $N=9.8*10^{26}atoms/m^{3}$

Radius of hydrogen atom, $R=0.53*10^{-10}m$

Permittivity of free space $\epsilon_{0}=8.85*10^{-12}Fm^{-1}$

Electronic polarizability, $\alpha _{e}=4\pi \epsilon_{0}(R)^{3}$

$\alpha _{e}=4*3.148(8.85*10^{-12}Fm^{-1})(0.53*10^{-10}m)^{3}$

$\alpha _{e}=16.548*10^{-42}Fm^{-2}$

Polarization is given by,  $P=\epsilon_{0}(\epsilon_{r}-1)E$  and $P=N\alpha _{e}E$

$N\alpha _{e}E=\epsilon_{0}(\epsilon_{r}-1)E$

Relative permittivity, $\epsilon_{r}=\frac{N\alpha _{e}}{\epsilon_{o}}+1$

$\epsilon_{r}=\frac{(9.8*10^{26}atoms. m^{-3})(16.548*10^{-42}Fm^{-2})}{8.854*10^{-12}Fm^{-1}} +1$

$\epsilon_{r}=18.316*10^{-4} +1$

$\epsilon_{r}=0.0018316+1$

$\epsilon_{r}=1.0018316$

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Answer 2

$16.5481042 Fm^{2}$ is the polarizability value.

$1.0018316$ is the value of relative permittivity.

Explanation :

The number of hydrogen atoms in a gas, $N = 9.8 * 10^{26} atoms/m^{3}$.

The radius of the hydrogen atom, $R = 0.53 * 10^{-10}m$.

The permittivity of free space, ∈$_{0} = 8.85*10^{-12} Fm^{-1}$.

Electronic polarizability, α$_{e} = 4\pi$∈$_{0} (R)^{3}$.

α$_{e} = 4*3.148(8.85*10^{-12} Fm^{-1} )(0.53*10^{-10} m)^{3}$

α$_{e} = 16.548 * 10^{-42} Fm^{-2}$

Polarization is given by, $P =$ ∈$_{0} ($∈$_{r} - 1) E$ and $P = N$α$_{e} E$

$N$α$_{e} E =$ ∈$_{0} ($∈$_{r} -1 ) E$

Relative permittivity, ∈$_{r} = N$α$_{e} /$∈$_{0} +1$

∈$_{r} = (9.8*10^{26} atoms.m^{-3} )(16.548*10^{-42} Fm^{-2} ) / 8.854 * 10^{-12} Fm^{-1} +1$

∈$_{r} = 18.316 * 10^{-14} + 1$

∈$_{r} = 0.0018316+ 1$

∈$_{r} = 1.0018316$

Therefore, the answer is  ∈$_{r} = 1.0018316$.

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