Quantum Chemistry

To determine the energy of an electron in the first excited state ( ) within a one-dimensional (1D) box of length , we use the quantum mechanical model for a particle in a box. The energy levels are quantized and given by the formula: where:

• : energy of the electron in the state
• : principal quantum number (for the first excited state, )
• : Planck's constant ( )
• : mass of the electron ( )
• : length of the box ( )

1. Substitute the known values into the equation:
   
2. Calculate the numerator:
   
3. Calculate the denominator:
   
4. Compute the energy :
   
5. Convert energy from joules to electron volts (1 eV = ):
   
6. Rounding to an appropriate number of significant figures:
   

Therefore, the energy of the electron in the first excited state is approximately 150.4 eV.

To determine the wavelength ( ) of a proton moving at a given velocity, we utilize the de Broglie hypothesis, which relates the wave-like and particle-like properties of matter. The de Broglie equation is given by: where:

• : wavelength of the particle
• : Planck’s constant ( )
• : mass of the particle
• : velocity of the particle

Given:

• Mass of proton ( ) =
• Velocity of proton ( ) =

1. Step 1: Substitute the Known Values into the de Broglie Equation
   
2. Step 2: Calculate the Denominator
   
3. Step 3: Compute the Wavelength ( )
   
4. Step 4: Convert Meters to Nanometers
   Since :
5. Step 5: Round to Appropriate Significant Figures
   Given the velocity is provided as (two significant figures), we round the wavelength accordingly:

Conclusion:
The wavelength associated with a proton moving at is approximately 0.40 nm. Therefore, the correct option is: