A certain dye absorbs 4000 Å and fluoresces at 8000 Å. These being wavelengths of maximum absorption that under given conditions 50 % of the absorbed energy is emitted. Calculate the ratio of the no. of quanta emitted to the number absorbed.

Answer

To solve the problem, we need to calculate the ratio of the number of quanta emitted (N_E) to the number of quanta absorbed (N_A) by a dye that absorbs a specific wavelength and fluoresces at another wavelength.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Wavelength of maximum absorption (λ_absorbed) = 4000 Å
- Wavelength of maximum emission (λ_emitted) = 8000 Å
- Percentage of absorbed energy emitted = 50%

2. **Use the Formula for Energy of a Photon:**
The energy (E) of a photon can be calculated using the formula:
\[ E = \frac{H \cdot C}{\lambda} \]
where:
- H = Planck's constant (6.626 x 10^-34 J·s)
- C = Speed of light (3 x 10^8 m/s)
- λ = Wavelength in meters

3. **Calculate the Energy of Absorbed and Emitted Photons:**
- Energy absorbed (E_A) at 4000 Å:
\[ E_A = \frac{H \cdot C}{\lambda_{absorbed}} = \frac{H \cdot C}{4000 \times 10^{-10} \text{ m}} \]
- Energy emitted (E_E) at 8000 Å:
\[ E_E = \frac{H \cdot C}{\lambda_{emitted}} = \frac{H \cdot C}{8000 \times 10^{-10} \text{ m}} \]

4. **Relate the Number of Quanta Emitted and Absorbed:**
Since 50% of the absorbed energy is emitted, we can express this relationship as:
\[ E_E = 0.5 \cdot E_A \]

5. **Substituting the Energy Values:**
\[ \frac{H \cdot C}{8000 \times 10^{-10}} = 0.5 \cdot \frac{H \cdot C}{4000 \times 10^{-10}} \]

6. **Canceling Common Terms:**
Since \(H\) and \(C\) are constants, they can be canceled out from both sides:
\[ \frac{1}{8000} = 0.5 \cdot \frac{1}{4000} \]

7. **Simplifying the Equation:**
Rearranging gives:
\[ \frac{1}{8000} = \frac{1}{8000} \]
This confirms that the relationship holds.

8. **Finding the Ratio of Quanta:**
The number of quanta emitted (N_E) and absorbed (N_A) can be expressed as:
\[ N_E = \frac{E_E}{E_{photon}} \quad \text{and} \quad N_A = \frac{E_A}{E_{photon}} \]
Thus, the ratio of the number of quanta emitted to the number absorbed is:
\[ \frac{N_E}{N_A} = \frac{E_E}{E_A} = \frac{0.5 \cdot E_A}{E_A} = 0.5 \]

9. **Final Ratio Calculation:**
Since we are looking for the ratio of quanta emitted to absorbed:
\[ \frac{N_E}{N_A} = \frac{1}{1} \]

### Conclusion:
The ratio of the number of quanta emitted to the number absorbed is **1:1**.