A negative hydrogen ion is a hydrogen atom containing two electrons. An electron and a negative hydrogen ion are accelerated through the same potential. Find the ratio of their speeds after acceleration, assuming the speeds are nonrelativistic. Use a value of 1.67 times 10 to the negative twenty-seventh kilograms for the mass of the hydrogen ion.
We have in this example a negative hydrogen ion. The way we would make such an ion is to take a neutral hydrogen atom, which has one proton and one electron, and add an electron to it so that overall it’s now a negative ion. In the situation described here, we can imagine two parallel plates some distance apart from one another. One plate has positive charge on it, and the other plate has a net negative charge. We can say then that some potential difference, we’ll call it 𝛥𝑉, exists between these two plates if we have an electron and a negative hydrogen ion begin at the negatively charged plate. We know that because of electrical repulsion and retraction, they’ll be pulled to the left towards the positive plate.
Once these two objects reach the other side, they’ll have accelerated through the same potential difference and they’ll have some final speed. It’s the ratio of those speeds that we want to solve for, where we’ll call the final speed of the electron 𝑉 sub e and the final speed of the hydrogen ion 𝑉 sub H minus. To start solving for this ratio, let’s consider this whole situation from an energy perspective. When we start out — that is, when the electron and the negative hydrogen ion are on the negative plate — they have no kinetic energy; they’re not in motion. But they have lots of potential energy. That’s because this negative plate is pushing them away while the positive plate is pulling them toward it. Soon this potential energy is converted into kinetic energy, energy of motion, as the electron and the hydrogen ion are accelerated across this gap.
Once they’ve reached the positive plate, we can say that both objects have no more electrical potential energy. They would prefer to stay just where they are. At a top level, we can write this when it comes to energy in this scenario. The potential energy that our objects start out with is equal to their final kinetic energy after being accelerated through the potential. And not only is this statement true overall, it’s also true individually for both of our objects. We can write that the potential energy of the electron initially is equal to its final kinetic energy and that the initial potential energy of our hydrogen ion is equal to its final kinetic energy. Let’s recall now just what electrical potential energy is as well as kinetic energy.
The electric potential energy of an object is equal to its electric potential in volts multiplied by the charge of that object. The idea then is we have some object with some total charge lowercase 𝑞 and that object is moved through a potential difference 𝑉. Well if we take the product of these two terms, that gives us the overall electric potential energy. If we apply this relationship to our two equations, we can rewrite the electric potential energy of the electron and the hydrogen ion. It’s equal to the potential difference each one went through, which in both cases is 𝛥𝑉 multiplied by the charge of the object overall, the charge of an electron and the charge of a negative hydrogen ion. We can recall the charge of a single electron 𝑞 sub e, but what about the charge of the negative hydrogen ion? What’s that?
Thinking back to our sketch of the negative hydrogen ion we see it has one proton and one electron whose charges cancel one another out and then one extra electron. Overall, this means that the charge of our negative hydrogen ion is the same as the charge on a single electron. That’s the net charge of this negative ion. So we can write 𝑞 sub e as the net charge of both of our objects. Now let’s consider the right hand side of these equations, the kinetic energy side. We can recall that the mathematical relationship for nonrelativistic kinetic energy is that it’s equal to the mass of the object in motion times one-half multiplied by the speed of the object squared. When we apply this relationship to our equations for energy balance, we can write out that the kinetic energy of the electron is one-half 𝑚 sub e times 𝑉 sub e squared and that of the negative hydrogen ion is one-half 𝑚 sub H minus times 𝑉 sub H minus squared.
We’ve been doing a lot of substituting and rearranging, but notice that we’re at a good place now. We want to solve for the ratio 𝑉 sub e to 𝑉 sub H minus. And in our two equations, we have a term 𝑉 sub e and a term 𝑉 sub H minus. But how will we create these two terms as a ratio, one divided by the other. One way we can do this is to divide our top equation by the bottom energy balance equation. We can do it this way because since these are two equations that means the left side and the right side are equal. And we see that when we do this division, a lot cancels out. For example, on the left side, 𝛥𝑉 cancels out as well as 𝑞 sub e. This whole left side reduces to one. Then on the right- hand side, we see that the factors of one-half cancel out. With this simplification, our mission should we choose to accept it is to rearrange this equation so we get the fraction 𝑉 sub e over 𝑉 sub H minus. That’s what we want to know.
We can start heading in this direction by multiplying both sides of the equation by the mass of the negative hydrogen ion divided by the mass of the electron. Looking at the right-hand side. we cancel out our electron mass as well as the ion mass. Look at the equation that we have now. All we need to do to find the ratio 𝑉 sub e over 𝑉 sub H is to take the square root of both sides of our equation. We’re now in the final stages and all we need to do to solve for the ratio is to enter in the values of the negative hydrogen ion mass as well as the mass of the electron and calculate this. We’re told that the mass of the negative hydrogen ion is 1.67 times 10 to the negative twenty-seventh kilograms, and the mass of an electron is a constant value we can look up in a table. A commonly accepted approximation for that mass is 9.1 times 10 to the negative thirty-first kilograms. So we have some very small masses here, and we’ll see how this ratio works out. When we enter in these masses, divide them, and take their square root, we find that the ratio is equal to 42.8. That is how many times faster the electron is moving than the negative hydrogen ion once they’ve been accelerated across the same potential.