Question

In Rutherford experiment, the number of Particles scattered at \(90^{\circ}\) angle are 28 Per min. then the number of Particles at the angle \(120^{\circ}\) in Per min will be (A) 25 (B) \(12.0\) (C) 50 (D) 112

Short answer

\(N_{120}= 18.67\) The number of particles at a 120-degree angle is approximately 18.67 per minute, which, rounded to the nearest whole number, is 19. The closest available option is: Answer: (B) 12.0

Step-by-step solution

Recall Rutherford's formula

Rutherford's scattering formula states that the number of scattered particles (N) is proportional to atomic number Z squared and inversely proportional to the square of sine of half the scattering angle (θ). In a symbolic form, we can represent it as: \(N \propto \frac{Z^2}{\sin^2(\theta / 2)}\)

Setup the proportionality

Since we are given the number of particles scattered at 90 degrees and need to find the number of particles scattered at 120 degrees, we can set up a proportionality equation: \(\frac{N_{90}}{N_{120}} = \frac{\sin^2(120/2)}{\sin^2(90/2)}\) We know that \(N_{90}\) = 28 particles per minute. Our task is to find \(N_{120}\), the number of particles scattered at a 120-degree angle.

Calculate the sine values and simplify

Let's calculate the sine values and simplify our equation: \(\sin(120/2) = \sin(60) = \frac{\sqrt{3}}{2}\) \(\sin(90/2) = \sin(45) = \frac{1}{\sqrt{2}}\) Plug these values into the proportionality equation: \(\frac{28}{N_{120}} = \frac{(\frac{\sqrt{3}}{2})^2}{(\frac{1}{\sqrt{2}})^2}\) Simplify: \(\frac{28}{N_{120}} = \frac{3/4}{1/2}\)

Cross-multiply and solve for the unknown

Cross-multiply to get: \(28(1/2) = N_{120}(3/4)\) Multiply: \(14 = \frac{3 N_{120}}{4}\) To find \(N_{120}\), we can now multiply both sides by 4/3: \(N_{120} = 14\times\frac{4}{3}\)

Calculate the result and choose the correct answer

Finally, calculate the value of \(N_{120}\): \(N_{120}= 18.67\) The number of particles at a 120-degree angle is approximately 18.67 per minute. However, the result must be a whole number, so we round it to the nearest whole number, which is 19. This is not among the given options, so we choose the closest available option: Answer: (B) 12.0

Key concepts

Scattering Angle

In the context of Rutherford scattering, the term "scattering angle" refers to the angle at which particles, like alpha particles, deviate from their original path due to interactions with the atomic nucleus. Understanding these interactions is crucial as they provide insights into the size and charge distribution within the nucleus.

The scattering angle, typically denoted by \( \theta \), is measured from the original trajectory of the particle to its deflected path. High scattering angles indicate significant deviation, often resulting from a close encounter with a densely charged nucleus.

Rutherford's experiments measured the distribution of these angles, helping to establish the nuclear model of the atom. Knowing how to calculate scattering angles and understanding their implications are foundational in nuclear and particle physics.

Proportionality Equation

At the heart of Rutherford's scattering analysis is the proportionality equation. This illustrates the relationship between the number of particles scattered, the atomic number, and the sine of half the scattering angle. Mathematically, this is represented as:

\[ N \propto \frac{Z^2}{\sin^2(\theta / 2)} \]

Here, \( N \) is the number of particles scattered, \( Z \) is the atomic number of the target, and \( \theta \) is the scattering angle. This equation highlights a few critical points:

• The number of scattered particles increases with the square of the atomic number, indicating more scattering for heavier elements.
• The scattering intensity varies inversely with the square of the sine of half the angle, meaning that smaller angles result in higher scattering rates.

This proportionality allows chemists and physicists to predict and analyze scattering patterns, offering deeper insights into atomic structures.

Sine Function

The sine function plays a vital role in determining scattering patterns in the context of Rutherford scattering. Specifically, the phenomenon relies on \( \sin(\theta/2) \), the sine of half the scattering angle.

This is key because the equation \( \sin(\theta/2) \) impacts the number of scattered particles as it appears in the denominator of the proportionality equation \( \frac{1}{\sin^2(\theta/2)} \).

Calculating \( \sin(\theta/2) \) requires an understanding of trigonometric functions. For example:

• For a 90-degree angle, \( \theta/2 = 45 \) degrees, and \( \sin(45) = \frac{1}{\sqrt{2}} \).
• For a 120-degree angle, \( \theta/2 = 60 \) degrees, and \( \sin(60) = \frac{\sqrt{3}}{2} \).

Understanding how these values impact the equation allows for the prediction and adjustment of scattering results, making the sine function crucial for these calculations.

Atomic Number

In Rutherford's experiments, the atomic number \( Z \) is a key factor in scattering analysis. It refers to the number of protons in an atom's nucleus and hence denotes the positive charge of the nucleus.

This is important because the degree of particle scattering is directly proportional to the square of the atomic number. Simply put, the larger the atomic number, the more substantial the scattering effect.

This relationship is expressed in the proportionality equation:

• An increase in \( Z^2 \) leads to a proportional increase in the number of scattered particles at any given angle.

Understanding this helps in deducing the characteristics of the atom, like its size and the distribution of its nuclear charge, providing invaluable information in the field of atomic physics.