An argon-ion laser emits a blue-green beam of light with a wavelength of 488 nm in a vacuum. What is the difference in energy in joules between the two energy states for the atomic transition that produces this light?

Options

A. `4.08xx10^(-19)J`
B. `6.18xx10^(-20)J`
C. `1.05xx10^(-20)J`
D. `4.76xx10^(-24)J`

Answer

To find the difference in energy in joules between the two energy states for the atomic transition that produces the blue-green light emitted by the argon-ion laser, we will use the formula for energy associated with a photon:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( E \) is the energy in joules,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^{8} \, \text{m/s} \)),
- \( \lambda \) is the wavelength in meters.

### Step 1: Convert the wavelength from nanometers to meters.
The given wavelength is \( 488 \, \text{nm} \).
To convert nanometers to meters, we use the conversion factor \( 1 \, \text{nm} = 10^{-9} \, \text{m} \):

\[ \lambda = 488 \, \text{nm} = 488 \times 10^{-9} \, \text{m} = 4.88 \times 10^{-7} \, \text{m} \]

### Step 2: Substitute the values into the energy formula.
Now we will substitute \( h \), \( c \), and \( \lambda \) into the energy formula:

\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3.00 \times 10^{8} \, \text{m/s})}{4.88 \times 10^{-7} \, \text{m}} \]

### Step 3: Calculate the energy.
Now we will perform the calculation:

\[ E = \frac{(6.626 \times 10^{-34}) \times (3.00 \times 10^{8})}{4.88 \times 10^{-7}} \]

Calculating the numerator:

\[ 6.626 \times 10^{-34} \times 3.00 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]

Now divide by the wavelength:

\[ E = \frac{1.9878 \times 10^{-25}}{4.88 \times 10^{-7}} \approx 4.07 \times 10^{-19} \, \text{J} \]

### Step 4: Final result.
The difference in energy between the two energy states for the atomic transition that produces this light is approximately:

\[ E \approx 4.07 \times 10^{-19} \, \text{J} \]