A mass spectrometer as shown in the attached figure is used to separate isotopes. If the bean emerges with a speed of 363 km/s and the magnetic field in the mass selector is 1.4 T, what is the distance between the collectors for 235U+1 and 238U+1? The mass of 235U+1 is 235 u and the mass of 238U+1 is 238 u, where the atomic mass unit u = 1.66 x 10-27 kg.
To solve this problem, we can use the principles of magnetic fields and the motion of charged particles in a magnetic field. Let's break it down step by step:
1. The first step is to determine the velocity of the charged particles in terms of their kinetic energy. We can use the formula KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
2. Since the kinetic energy is given in terms of the velocity, we can rearrange the formula to solve for v: v = √((2KE)/m).
3. Next, we convert the velocity from km/s to m/s by multiplying it by 1000 since there are 1000 meters in a kilometer.
4. We can then calculate the velocity of the charged particles using the given kinetic energy and the mass of each isotope.
5. The next step is to determine the radius of the circular path of the charged particles in the magnetic field. We can use the formula r = (mv)/(qB), where r is the radius, m is the mass, v is the velocity, q is the charge of the particle, and B is the magnetic field strength.
6. Using the given magnetic field strength and the calculated velocity, we can calculate the radius of the path for each isotope.
7. Finally, we find the distance between the collectors by subtracting the radius of the path for 238U+1 from the radius of the path for 235U+1.
Let's plug in the values and calculate the answer:
Given:
Kinetic energy (KE) = 1/2 mv^2
Mass of 235U+1 (m1) = 235 u
Mass of 238U+1 (m2) = 238 u
Atomic mass unit (u) = 1.66 x 10^-27 kg
Velocity (v) = 363 km/s = 363,000 m/s
Magnetic field (B) = 1.4 T
Step 1:
Calculate the velocity using kinetic energy:
v = √((2KE)/m)
Step 2:
Convert the velocity from km/s to m/s:
v = 363,000 m/s
Step 3:
Calculate the radius using the formula r = (mv)/(qB):
r1 = (m1v)/(qB)
r2 = (m2v)/(qB)
Step 4:
Calculate the distance between the collectors by subtracting the radii:
Distance = r2 - r1
By following these steps and plugging in the given values, you should be able to find the distance between the collectors for 235U+1 and 238U+1 in the mass spectrometer.