Three emission lines involving three energy levels in an atom occur at wavelengths x, 1.5x, and 3.0x nanometers. Which wavelength corresponds to the transition from the highest to the lowest of the three energy levels?
Please help me, thank you very much for your time.
delta E = hc/lambda. Going from the highest to the lowest energy level must involve the highest E. To have the highest E, lambda must be the ??
is it the wavelength? I have no idea.
C'mon now. You need to understand how to do this and this is a simple problem in logic. Here is the way the logic goes. Suppose we have an equation that is x = 5/y. Now, as y gets larger, x must get smaller. Try it. y = 1 then x = 5/1 = 5, right? y = 2, then x = 5/2 = 2.5 (x is smaller when y is larger), right? y = 5, then x = 5/5 = 1 (smaller than when y was larger). Now apply that same reasoning to
E = hc/lambda. h is a constant, c is a constant, the wavelength is the only variable. You want delta E to be the largest it can be because the transition is from the highest to the lowest level which means highest E. So lambda must be?? (smaller or larger are the only two choices and that dictates the answer of x, 1.5x or 3.0x.)
To determine which wavelength corresponds to the transition from the highest to the lowest of the three energy levels, we need to understand the relationship between energy levels and wavelengths in atomic emission.
In atomic emission, when an electron transitions from a higher energy level to a lower energy level, it emits a photon of light with a specific wavelength. The energy difference between the levels determines the wavelength of the emitted light.
The formula that relates wavelength (λ) to energy level is given by the Rydberg formula:
1/λ = R * (1/n₁² - 1/n₂²)
where R is the Rydberg constant (approximately 1.097 x 10^7 1/m), and n₁ and n₂ are the principal quantum numbers corresponding to the initial and final energy levels, respectively.
In this case, we have three energy levels and three corresponding wavelengths: x, 1.5x, and 3.0x nanometers.
Let's assume that the highest energy level corresponds to n₃ and the lowest energy level corresponds to n₁. We want to find the wavelength corresponding to the transition from n₃ to n₁.
We can set up two equations using the Rydberg formula:
1/λ₁ = R * (1/n₃² - 1/n₂²) and
1/λ₂ = R * (1/n₃² - 1/n₁²)
Substituting the given wavelengths into the equations:
1/x = R * (1/n₃² - 1/n₂²)
1/1.5x = R * (1/n₃² - 1/n₁²)
We can solve these equations simultaneously to find the values of n₃ and n₁. Once we have those values, we can determine the wavelength (corresponding to the transition from the highest to the lowest energy level) using the Rydberg formula.
Alternatively, we can solve these equations algebraically to eliminate the unknowns n₃ and n₁. We can divide the two equations and rearrange the terms:
(1/x) / (1/1.5x) = (R * (1/n₃² - 1/n₂²)) / (R * (1/n₃² - 1/n₁²))
Simplifying:
1.5 = (n₃² - n₂²) / (n₃² - n₁²)
1.5(n₃² - n₁²) = (n₃² - n₂²)
Expanding and rearranging:
1.5n₃² - 1.5n₁² = n₃² - n₂²
(0.5n₃² - 1.5n₁²) = -n₂²
Substituting n₁=x, n₂=1.5x, and n₃=3x:
0.5(9x²) - 1.5(x²) = -2.25x² + 1.5x² = -0.75x²
From this equation, we can see that the wavelength corresponding to the transition from the highest to the lowest energy level is at -0.75x² nanometers.
Therefore, the negative value suggests that there might be an error in the provided data or in our calculations. It's crucial to double-check the information to ensure its accuracy.
Remember, the steps above explain the process of obtaining the solution by using the Rydberg formula and algebraic manipulation. However, in practice, it is important to verify the provided data and perform the necessary calculations accurately.