Solution:

Introduction:
Intrinsic conductivity is defined as the conductivity of a material in its purest form, without any impurities. It is denoted by σi and is given by the product of intrinsic carrier concentration (ni) and mobility of free electrons or holes (μn or μp).

σi = ni * μn (for electrons) or σi = ni * μp (for holes)

Given:
Mobility of free electrons in silicon (μn) = 0.13 m²/V-s
Mobility of holes in silicon (μp) = 0.05 m²/V-s
Intrinsic carrier concentration of silicon (ni) = 1.5×10^16/m³
Mobility of free electrons in germanium (μn) = 0.38 m²/V-s
Mobility of holes in germanium (μp) = 0.18 m²/V-s
Intrinsic carrier concentration of germanium (ni) = 2.5×10^19/m³

Calculations:

For Silicon:
σi = ni * μn + ni * μp
σi = 1.5×10^16/m³ * 0.13 m²/V-s + 1.5×10^16/m³ * 0.05 m²/V-s
σi = 2.25 S/m

Therefore, the intrinsic conductivity of silicon is 2.25 S/m.

For Germanium:
σi = ni * μn + ni * μp
σi = 2.5×10^19/m³ * 0.38 m²/V-s + 2.5×10^19/m³ * 0.18 m²/V-s
σi = 9.5 S/m

Therefore, the intrinsic conductivity of germanium is 9.5 S/m.

Explanation:
The intrinsic conductivity of a material depends on two factors: intrinsic carrier concentration (ni) and mobility of free electrons or holes (μn or μp). Intrinsic carrier concentration is the concentration of electrons and holes in a material when no impurities are present. It depends on the band gap of the material and the temperature. The mobility of free electrons and holes is the ease with which they can move in the crystal lattice of the material. It depends on the crystal structure, temperature, and impurities present in the material.

In the given problem, we are given the values of mobility of free electrons and holes and intrinsic carrier concentration for silicon and germanium. We use the formula for intrinsic conductivity to calculate the intrinsic conductivity of both materials. We can see that the intrinsic conductivity of germanium is higher than that of silicon. This is because germanium has a smaller band gap than silicon, which means that it has more free electrons and holes at room temperature. Also, the mobility of free electrons and holes is higher in germanium than in silicon, which makes it easier for them to move around in the crystal lattice. As a result, germanium has a higher intrinsic conductivity than silicon.