Ok, let's use the following information to very roughly estimate the size of the nucleus. Rutherford, Geiger, and Marsden (his students) fired a beam of alpha particles at a gold foil. The alpha particles had an energy of about 3.70 million electron volts, a charge of +2e, and a mass of about four AMU (one Atomic Mass Unit is 1.66 x 10-27 kg. Gold has atomic number 79.
How close could the alpha particles get to the gold nucleus?
To estimate how close the alpha particles could get to the gold nucleus, we can use Rutherford's scattering formula. This formula relates the scattering angle of the alpha particles to the impact parameter, which is the distance of closest approach between the alpha particle and the gold nucleus.
The formula is given as:
θ = 2 * arctan(b / D)
where:
- θ is the scattering angle
- b is the impact parameter
- D is the distance between the source and the detector of the scattered particles
In Rutherford's experiment, the maximum scattering angle observed was about 180 degrees, which indicates that the alpha particles made a U-turn after passing close to the nucleus.
Using this information, we can assume that the distance of closest approach corresponds to the distance at which the alpha particle had a scattering angle of 90 degrees.
Let's calculate the maximum impact parameter (b_max) using the given information:
θ = 90 degrees = π/2 radians
D = Not provided in the information given.
Since D is not provided in the information, we cannot estimate the exact distance that the alpha particles traveled before reaching the gold foil. However, we can use a rough approximation by assuming that the alpha particles had a straight trajectory before reaching the foil.
Now, let's assume that the distance between the source and the foil (D) is about 1 meter (m) for illustrative purposes.
θ = 90 degrees = π/2 radians
D = 1 m
Plugging these values into the formula, we can find the maximum impact parameter:
π/2 = 2 * arctan(b_max / 1)
π/4 = arctan(b_max)
Taking the tangent of both sides, we get:
tan(π/4) = tan(arctan(b_max))
1 = b_max
Therefore, the maximum impact parameter (closest distance of approach) for the alpha particles to the gold nucleus, with this rough estimate, is approximately 1 meter.
Please note that this estimate is very rough and not based on the specific parameters of Rutherford's experiment.
To estimate how close the alpha particles could get to the gold nucleus, we can use the principles of Rutherford's scattering experiment and the kinetic energy of the alpha particles.
In Rutherford's experiment, he observed that most of the alpha particles passed through the gold foil, but a small fraction experienced large-angle scattering or even bounced back. This led him to propose that the positive charge and most of the mass of an atom is concentrated in a small, dense region called the nucleus.
The kinetic energy (KE) of the alpha particles can be calculated using the formula: KE = (1/2)mv^2, where m is the mass and v is the velocity of the alpha particle.
Given:
- Energy of alpha particles = 3.70 million electron volts
- Charge of alpha particles = +2e (where e is the elementary charge, 1.60 x 10^-19 coulombs)
- Mass of alpha particles = 4 AMU (1.66 x 10^-27 kg per AMU)
First, we need to convert the energy of the alpha particles from electron volts to joules. 1 electron volt is equal to 1.60 x 10^-19 joules.
Energy in joules = 3.70 million electron volts * 1.60 x 10^-19 joules per electron volt.
Next, we can calculate the velocity of the alpha particles using the equation KE = (1/2)mv^2. Rearranging the equation, we get v = √(2KE/m).
After calculating the velocity, we can find the distance of closest approach (r) between the alpha particle and the gold nucleus using the equation for the Coulomb force: F = (k * (q1 * q2)) / r^2, where F is the force, k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the distance.
Since the alpha particles are positively charged (+2e), and the gold nucleus is positively charged with atomic number 79, we can substitute the values into the equation as follows: F = (k * (2e * 79e)) / r^2.
Now, let's calculate the distance of closest approach using these steps.