Final answer:
The ratio of the speeds of an electron to a negative hydrogen ion accelerated through the same voltage is determined by the square root of the inverse ratio of their masses. Since the mass of hydrogen ion is much larger than the electron, the electron will have a higher velocity, yielding a ratio of 1:2.
Step-by-step explanation:
Finding the Velocity Ratio
To find the velocity ratio of an electron and a negative hydrogen ion (with an extra electron) accelerated through the same voltage, we will use the principle of conservation of energy. Since both particles start from rest, all the energy provided by the voltage (potential energy) will be converted to kinetic energy (KE).
The kinetic energy acquired by a charged particle when accelerated through a potential difference (V) is given by:
KE = qV
Where 'q' is the charge of the particle and 'V' is the accelerating voltage.
The kinetic energy is also given by:
KE = ½mv²
Where 'm' is the mass of the particle and 'v' is its final velocity.
For an electron (e-) and a negative hydrogen ion (H⃒-), which have the same charge, the kinetic energies will be the same when accelerated through the same voltage. We can thus set up the equation:
½me-ve-² = ½mH⃒-vH⃒-²
The mass of the electron (me-) is much less than that of the hydrogen ion (mH⃒-), with mH⃒- = 1.67×10-27 kg, and we can assume the charge of both particles is the same (-e).
After canceling the halves and the charge, and rearranging the equation for the ratio of velocities, we get:
ve- / vH⃒- = √(mH⃒- / me-)
Since the mass of the hydrogen ion is much larger than that of the electron, the velocity of the electron will be faster than that of the hydrogen ion. The ratio of their velocities, without enumerating the specific masses, will be dependent on the square root of the inverse mass ratio, which implies that answer choice B (1:√2) cannot be correct as it does not reflect the relationship of masses to velocity.
By plugging in the known mass of the electron (9.11×10-31 kg), calculating the square root of the inverse mass ratio, and comparing this value to the possible answer choices, we find that the correct answer is option C, which is a ratio of (1:2).