Problem 69 - Chemistry Central Science

Question

True or false: (a) The band gap of a semiconductor decreases as the particle size decreases, in the \(1-10-\mathrm{nm}\) range. (b) The light that is emitted from a semiconductor, upon external stimulation, is longer and longer in wavelength as the particle size of the semiconductor decreases.

Step-by-Step Solution

Answer

(a) False: As the particle size decreases, the band gap of a semiconductor increases. (b) False: As the particle size of a semiconductor decreases, the wavelength of the light emitted from it also decreases.

Step 1: Statement (a) Analysis

The statement (a) claims that the band gap of a semiconductor decreases as the particle size decreases in the range of 1-10 nm. The band gap of a semiconductor is the energy difference between the valence band and conduction band. This relationship is given by the following formula: \[E_g = \frac{h^2}{8m^*e^2}\frac{1}{r_0^2}\] where \(E_g\) is the band gap, \(h\) is Planck's constant, \(m^*\) is the effective mass of the electron, \(e\) is the charge of an electron, and \(r_0\) is the particle radius. This shows that the band gap of a semiconductor is inversely proportional to the square of the particle size. Thus, as the particle size decreases, the band gap increases. Therefore, the statement (a) is False.

Step 2: Statement (b) Analysis

The statement (b) claims that the wavelength of light emitted from a semiconductor gets longer as the particle size decreases. The energy of the emitted light is related to the band gap of the semiconductor. The relationship between the wavelength (\(\lambda\)) and energy (\(E_g\)) is given by the following formula: \[E_g = \frac{hc}{\lambda}\] where \(h\) is Planck's constant, \(c\) is the speed of light in vacuum. Since from statement (a), we know that the band gap increases as the particle size decreases, we can look at the formula for energy to determine the relationship of wavelength to particle size. As the band gap increases, it means the energy of light emitted increases, and therefore by the equation given above, the wavelength of the light emitted decreases. Thus, the statement (b) is False.