Question 36: Atoms & Nuclei

Question

If an electron in a hydrogen atom jumps from the third orbit to the second orbit, it emits a photon of wavelength $$\lambda$$. When it jumps from the second to the first orbit, the corresponding wavelength of the photon will be

Accepted Answer

To find the magnitude of the angular momentum of the electron, we can use the formula for the angular momentum of a particle moving in a circular orbit:
$$L = mvr$$
where:
$$m$$ is the mass of the electron, $$9.1 	imes 10^{-31} \mathrm{~kg}$$
$$v$$ is the speed of the electron, $$2.2 	imes 10^6 \mathrm{~ms}^{-1}$$
$$r$$ is the radius of the orbit, $$0.529 	imes 10^{-10} \mathrm{~m}$$
Now, let's substitute these values into the formula:
$$L = 9.1 	imes 10^{-31} \mathrm{~kg} 	imes 2.2 	imes 10^6 \mathrm{~ms}^{-1} 	imes 0.529 	imes 10^{-10} \mathrm{~m}$$
First, we will calculate the product: $2.2 	imes 10^6 	imes 0.529 	imes 10^{-10}$
$$2.2 	imes 0.529 = 1.1638$$
Combining the exponents of 10:
$$10^6 	imes 10^{-10} = 10^{-4}$$
So the product is:
$$1.1638 	imes 10^{-4}$$
Now calculate:
$$9.1 	imes 10^{-31} \mathrm{~kg} 	imes 1.1638 	imes 10^{-4}$$
Calculating the product of the coefficients: $9.1 	imes 1.1638$
$$9.1 	imes 1.1638 = 10.57758$$
Combining this with the exponent of 10:
$$10.57758 	imes 10^{-31} 	imes 10^{-4} = 10.57758 	imes 10^{-35}$$
We express this final value in scientific notation:
$$1.057758 	imes 10^{-34} \mathrm{Kg} \mathrm{m}^2 \mathrm{~s}^{-1}$$
Rounding this to three significant figures:
$$1.06 	imes 10^{-34} \mathrm{Kg} \mathrm{m}^2 \mathrm{~s}^{-1}$$
Therefore, the correct answer is:
Option A: $$1.06 	imes 10^{-34} \mathrm{Kg} \mathrm{m}^2 \mathrm{~s}^{-1}$$

Answer

$$
\frac{5 \lambda}{27}
$$

Answer

$$
\frac{7 \lambda}{20}
$$

Answer

$$
\frac{16 \lambda}{9}
$$

Answer

$$
\frac{20 \lambda}{7}
$$

Atoms & Nuclei

If an electron in a hydrogen atom jumps from the third orbit to the second orbit, it emits a photon of wavelength $\lambda$. When it jumps from the second to the first orbit, the corresponding wavelength of the photon will be

Choose the correct answer: