Question

Indium(III) phosphide is a semiconducting material that has been frequently used in lasers, light-emitting diodes (LED), and fiber-optic devices. This material can be synthesized at \(900 . \mathrm{K}\) according to the following reaction: $$ \operatorname{In}\left(\mathrm{CH}_{3}\right)_{3}(g)+\mathrm{PH}_{3}(g) \longrightarrow \operatorname{InP}(s)+3 \mathrm{CH}_{4}(g) $$ a. If 2.56 \(\mathrm{L} \operatorname{In}\left(\mathrm{CH}_{3}\right)_{3}\) at 2.00 \(\mathrm{atm}\) is allowed to react with 1.38 \(\mathrm{L} \mathrm{PH}_{3}\) at \(3.00 \mathrm{atm},\) what mass of InP(s) will be produced assuming the reaction has an 87\(\%\) yield? b. When an electric current is passed through an optoelectronic device containing InP, the light emitted has an energy of \(2.03 \times 10^{-19} \mathrm{J}\) . What is the wavelength of this light and is it visible to the human eye? c. The semiconducting properties of InP can be altered by doping. If a small number of phosphorus atoms are replaced by atoms with an electron configuration of \([\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4},\) is this n-type or p-type doping?

Short answer

a. The mass of InP produced with an 87% yield is 55.2 g. b. The emitted light has a wavelength of 975 nm, which is not visible to the human eye (infrared region). c. The doping process results in n-type semiconducting InP.

Step-by-step solution

Part a: Calculating the mass of InP produced

Given variables: \(V_{In(CH_3)_3} = 2.56 \, L\) \(P_{In(CH_3)_3} = 2.00 \, atm\) \(V_{PH_3} = 1.38 \, L\) \(P_{PH_3} = 3.00 \, atm\) \(T = 900 \, K\) Reaction yield: \(87\%\) 1. Calculate moles of In(CH\(_3\)\(_3\) and PH\(_3\) using the Ideal Gas Law (\(PV = nRT\)). \( n_{In(CH_3)_3} = \frac{P_{In(CH_3)_3} V_{In(CH_3)_3}}{R T}\) \( n_{PH_3} = \frac{P_{PH_3} V_{PH_3}}{R T}\) 2. Compare the mole ratio to determine the limiting reactant. Observe the stoichiometry of the balanced equation and compare the moles calculated from the given initial quantities. 3. Calculate the moles of InP formed based on the limiting reactant and reaction yield (87%). Multiply the moles of the limiting reactant by the reaction yield and the stoichiometric ratio. 4. Calculate the mass of InP produced. Using the molar mass of InP, convert the moles of InP formed to mass.

Part b: Calculating the wavelength of emitted light

Given variables: Energy of the emitted light: \(2.03 \times 10^{-19} \, J\) 1. Use the Planck's equation to find the wavelength of the emitted light. \(E = h f\), and \(f = \frac{c}{\lambda}\), therefore, \(E = \frac{hc}{\lambda}\). 2. Calculate the wavelength in nanometers: Solve for the wavelength, \(\lambda\), and convert it to nanometers. 3. Determine if the wavelength is visible to the human eye. Based on the visible light spectrum (approx 380 nm to 740 nm), decide if the calculated wavelength is visible or not.

Part c: Identifying the doping process

Given electron configuration for the doping metal atom: \([Kr] 5s^2 4d^{10} 5p^4\) 1. Identify the group of the doping metal in the periodic table: Analyze the electron configuration provided to determine the doping metal's group number in the periodic table. 2. Compare the number of valence electrons of the doping metal with phosphorus: Compare the valence electron count of the doping metal with that of phosphorus atoms in InP. 3. Determine if it's n-type or p-type doping: Based on the comparison made, identify if the doping results in n-type or p-type semiconducting InP.

Key concepts

Semiconductors

Semiconductors are materials that have a conductivity between that of a conductor (like copper) and an insulator (such as glass). These materials are the backbone of modern electronics and play a crucial role in devices like transistors, diodes, and solar cells. Indium phosphide (InP) is a well-known semiconductor.

• Characteristic: Semiconductors have a unique band gap that allows them to conduct electricity under certain conditions, which makes them essential in controlling electrical flow in circuits.
• Usage: They are widely used in optoelectronic devices, such as LEDs and laser diodes, because they can efficiently convert electrical energy into light.

Semiconductors are often used in temperature measurements because their resistance changes with temperature, providing precise readings. This unique property underscores their critical role in both everyday electronics and advanced scientific equipment.

Ideal Gas Law

The Ideal Gas Law, represented by the equation \(PV = nRT\), is a fundamental principle used to describe the behavior of gases. It allows us to relate the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of a gas to the number of moles (\(n\)) present.

• Formula Components: \(R\) is the universal gas constant, commonly expressed as \(8.314 \, J/(mol\cdot K)\), which connects the macroscopic properties of gases.
• Application: In the given exercise, this law helps in calculating the moles of gaseous reactants for Indium phosphide synthesis at a specific temperature of \(900 \, K\), essential for predicting the amount of product formed.
• Real-World Example: The Ideal Gas Law also applies to air bags, where rapid inflation is achieved by decomposing sodium azide to produce gas, adhering closely to ideal gas principles.

Understanding the Ideal Gas Law is vital for chemists as it simplifies complex gas behaviors into tangible, calculable metrics.

Planck's Equation

Planck's equation links the energy of electromagnetic waves to their frequency through the relation \(E = hf\). Here, \(E\) represents energy, \(h\) is the Planck constant \(6.626 \,\times 10^{-34} \, J\cdot s\), and \(f\) is the frequency of the wave.

• Equation Manipulation: The equation \(E = hf\), and knowing \(f = \frac{c}{\lambda}\), leads us to \(E = \frac{hc}{\lambda}\), which is used to determine the wavelength of emitted light.
• Usage in Optoelectronics: It is applied to calculate the wavelength from the energy provided by devices like LEDs using materials such as Indium phosphide.
• Visible Light Determination: In the exercise, you calculate whether the wavelength (and therefore color) of emitted light falls within the visible spectrum (380 nm - 740 nm).

Planck's equation serves as a fundamental tool in understanding how energy scales with the properties of light, crucial for designing efficient optoelectronic components.

Doping in Semiconductors

Doping involves adding small quantities of impurities to a semiconductor to change its electrical conductivity. This process enhances the material properties of indium phosphide, fine-tuning it for specific electronic applications.

• P-Type Doping: Achieved by adding elements with fewer electrons than the host semiconductor, creating 'holes' or positive charge carriers.
• N-Type Doping: Involves adding elements that have more electrons than the semiconductor, thus increasing the number of negative charge carriers.
• Specific Example: In the given exercise, if phosphorus in InP is replaced by an atom with the configuration \( [Kr] 5s^2 4d^{10} 5p^4 \), it indicates a shift to p-type doping due to an excess of valence electrons.

Doping is essential because it allows control over the charge carrier density in semiconductors, dramatically altering their electronic properties and enabling their wide range of uses in circuitry and devices.