Question

Identify the INCORRECT statements from the following:

A. Notation \( {}^{24}_{12}\mathrm{Mg} \) represents 24 protons and 12 neutrons.
B. Wavelength of a radiation of frequency \( 4.5 \times 10^{15}\ \text{s}^{-1} \) is \( 6.7 \times 10^{-8}\ \text{m} \).
C. One radiation has wavelength \( \lambda_1 \) (900 nm) and energy \( E_1 \). Other radiation has wavelength \( \lambda_2 \) (300 nm) and energy \( E_2 \). \( E_1 : E_2 = 3 : 1 \).
D. Number of photons of light of wavelength 2000 pm that provides 1 J of energy is \( 1.006 \times 10^{16} \).

Choose the correct answer from the options given below: 

• A. A and D Only
• B. A and C Only
• C. A and B Only
• D. B and C Only

Hint:

Remember the relationships between Z, A, protons, and neutrons. Energy and wavelength are inversely related for photons.

Answer 1

The Correct Option is B

Let's analyze the statements one by one to determine which ones are incorrect.

1. Statement A: Notation ^{24}_{12}\text{Mg} represents 24 protons and 12 neutrons.
   • An element's notation ^{A}_{Z}\text{X} signifies that it has mass number A and atomic number Z.
   • For ^{24}_{12}\text{Mg}, A = 24 and Z = 12, which means it has 12 protons (equal to Z) and 24 - 12 = 12 neutrons.
   • Thus, statement A is incorrect because it incorrectly states that there are 24 protons and 12 neutrons.
2. Statement B: Wavelength of a radiation of frequency 4.5 \times 10^{15} \,\text{s}^{-1} is 6.7 \times 10^{-8} \,\text{m}.
   • Wavelength \lambda is related to frequency f by the formula: \lambda = \frac{c}{f}, where c = 3 \times 10^8 \,\text{m/s} is the speed of light.
   • Calculating the wavelength: \lambda = \frac{3 \times 10^8}{4.5 \times 10^{15}} \approx 6.67 \times 10^{-8} \,\text{m}.
   • The statement seems correct, as it closely matches the given value.
3. Statement C: One radiation has wavelength \lambda_1 (900 nm) and energy E_1. Another radiation has wavelength \lambda_2 (300 nm) and energy E_2. E_1 : E_2 = 3 : 1.
   • Energy of a photon is given by E = \frac{hc}{\lambda}.
   • For comparison: \frac{E_1}{E_2} = \frac{\lambda_2}{\lambda_1}.
   • Since \lambda_2 = 300 \,\text{nm} and \lambda_1 = 900 \,\text{nm}, we calculate:
   • \frac{E_1}{E_2} = \frac{300}{900} = \frac{1}{3}, whereas the statement claims it to be 3:1.
   • Thus, this statement is incorrect.
4. Statement D: Number of photons of light with wavelength 2000 pm that provides 1 J of energy is 1.006 \times 10^{16}.
   • Convert wavelength to meters: 2000 \,\text{pm} = 2000 \times 10^{-12} \,\text{m} = 2 \times 10^{-9} \,\text{m}.
   • Energy of one photon: E_{\text{photon}} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{2 \times 10^{-9}} \approx 9.939 \times 10^{-17} \,\text{J/photon}.
   • Number of photons for 1 J: \frac{1}{9.939 \times 10^{-17}} \approx 1.006 \times 10^{16}.
   • This statement is correct.

Therefore, the incorrect statements are A and C, making the correct answer: A and C Only.