In a salt crystal, the distance
between adjacent sodium and chloride ions is 3.42 times 10 to the negative 10th
metres. What is the force of attraction
between the two singly charged ions?
Considering this question, one way
to write out salt is to represent it using the symbols of each of its elements in
the periodic table, that is, sodium, which is represented by capital N and a, and
then chlorine, which makes up the chloride ion, which is represented as capital
Cl. We’re told that, in a salt crystal,
the distance separating the sodium and chloride ions, which we can refer to using
the letter 𝑑, is given as 3.42 times 10 to the negative 10th metres.
Knowing this, we want to solve for
the force of attraction between these two ions. And we’re also told that the ions
are charged, in particular singly charged, which we’ll talk about in a moment. The fact that we’re solving for the
force between two charged particles gives us a clue about what mathematical
relationship we’ll use to do this.
Coulomb’s law is a law for the
electrical force of attraction or repulsion between two charged particles with
charges 𝑄 one and 𝑄 two. The law says that if we take the
product of these two charges and then multiply them by something called Coulomb’s
constant, represented with a lowercase 𝑘, and then divide all that by the distance
between the charges squared, then, in that case, we’ve solved for the electrical
force between the charges.
In our case, we can write that
electric force, 𝐹 sub 𝑒, which by the way is the force of attraction we want to
solve for, as 𝑘 times 𝑄 one, the charge of our sodium ion, multiplied by 𝑄 two,
the charge of our chloride ion, all divided by the distance 𝑑 between the ions
squared. As we said, 𝑘 Coulomb’s constant
is indeed a constant. And the value we’ll use for it is
8.99 times 10 to the ninth newton metres squared per coulomb squared. Those are some complicated looking
units. But we’ll see why they are what
they are in a moment.
Next, let’s consider the two
charges of our ions, 𝑄 one and 𝑄 two. In the problem statement, we’re
told that the ions are singly charged. That means that the net charge on
each of our two ions, the sodium and chloride ions, is equal to one unit of the
smallest possible amount of electric charge, which is the charge of an electron or
the charge of a proton. Those charges have opposite signs
but the same magnitude, which forms our minimum value for charge.
And what is that minimum value? We can call it 𝑄. And it’s equal to 1.6 times 10 to
the negative 19th coulombs. That’s the charge of a proton. Or we could also call it the
magnitude of the charge of an electron. The fact that these two ions are
singly charged means that each of them has a net charge equal to lowercase 𝑞. That is, 𝑄 one and 𝑄 two are both
equal to 1.6 times 10 to the negative 19th coulombs.
Knowing all that, we are now ready
to plug in values to solve for 𝐹 sub 𝑒, the force of attraction between these
ions. When we insert these values, take a
look for a moment at the units. We mentioned how strange the units
of Coulomb’s constant 𝑘 appear. But we see now that they make
sense. In the context of Coulomb’s law,
the units of coulombs cancel out as do the units of metres squared. This means that, thanks to the
units of the constant 𝑘, our overall outcome will have units of newtons, the units
And entering this equation on our
calculator, we find a result of 1.97 times 10 to the negative ninth newtons. That is equal to the force of
attraction between the two singly charged ions.