Introduction:
In this problem, we are given a one-dimensional infinite potential well with a potential energy of zero between -a and a, and infinite outside this range. An electron is transitioning from the n=5 level to the n=2 level, and the frequency of the emitted photon is provided. Our goal is to find the width of the box, or the value of 'a'.

Plan:
To solve this problem, we need to apply the energy conservation principle. We know that the energy of the emitted photon is equal to the difference in energy between the initial and final states of the electron. Using the energy level formula for a one-dimensional infinite potential well, we can calculate the energy difference and then find the value of 'a'.

Calculations:
1. Energy Levels in a One-Dimensional Infinite Potential Well:
The energy levels of a particle in a one-dimensional infinite potential well are given by the equation: E_n = (n^2 * h^2) / (8mL^2), where n is the energy level, h is the Planck's constant, m is the mass of the particle, and L is the width of the well.

2. Energy Difference:
The energy difference between the n=5 and n=2 levels is given by ΔE = E_2 - E_5. Substituting the energy level formula, we have: ΔE = [(2^2 * h^2) / (8mL^2)] - [(5^2 * h^2) / (8mL^2)].

3. Frequency of the Emitted Photon:
Using the relation E = hf, where E is the energy of the photon and f is its frequency, we can write: ΔE = hf.

4. Solving for 'a':
Setting the energy difference ΔE equal to hf, we can equate the two expressions obtained in step 2 and 3. Rearranging and solving for 'a', we get: a = [(h * √(3f)) / (2√(10)m)].

Answer:
The width of the box, or the value of 'a', is given by a = [(h * √(3f)) / (2√(10)m)]. Substituting the given values of Planck's constant (h) and the frequency of the emitted photon (f), along with the mass of the electron (m), we can calculate the numerical value of 'a'.