Uranium 238 nucleus, initially at rest, emits an alpha particle with the speed of 1.4 × 10⁷ meters per second. Calculate the recoil speed of the residual nucleus. Thorium 234 assumes that the mass of the nucleus is proportional to the mass number.

To calculate the recoil speed of the residual nucleus (Thorium-234) after the emission of an alpha particle from Uranium-238, we can use the principle of conservation of momentum.

First, we need to determine the initial and final momenta of the system. Let's denote the initial mass of the Uranium-238 nucleus as M1, the initial velocity of the alpha particle as v_alpha, the final mass of the Thorium-234 nucleus as M2, and the recoil velocity of the Thorium-234 nucleus as v_thorium.

Conservation of mass:
Since mass is proportional to the mass number, we can write:
M1 = 238 * A
M2 = 234 * A
where A is a constant of proportionality.

Conservation of momentum:
Initial momentum = Final momentum
M1 * 0 (initial velocity of Uranium-238) = M2 * v_thorium (recoil velocity of Thorium-234) + m_alpha * v_alpha (momentum of the alpha particle)

Now, we can substitute the expressions for M1 and M2 using the constant of proportionality A:

238A * 0 = 234A * v_thorium + (4 * A) * v_alpha

Simplifying the equation:

0 = 234A * v_thorium + 4A * v_alpha

Now, we can solve for v_thorium:

234A * v_thorium = -4A * v_alpha

v_thorium = (-4A * v_alpha) / (234A)

The constant of proportionality A cancels out:

v_thorium = (-4 * v_alpha) / 234

Now, we can plug in the values:

v_alpha = 1.4 x 10^7 m/s
v_thorium = (-4 * 1.4 x 10^7 m/s) / 234

v_thorium ≈ -2.38 x 10^5 m/s

The negative sign indicates that the Thorium-234 nucleus moves in the opposite direction of the emitted alpha particle. So, the recoil speed of the Thorium-234 nucleus is approximately 2.38 x 10^5 m/s in the opposite direction to the alpha particle's motion.