The basic problem
The following types of expressions are ubiquitous, especially (but not only) in scientific and technical literature; note the indefinite article in the construction "…has/with a value of…":
At period 1, inflation has a value of about 1.021.
The electric field within the capacitor has a value of 170 N/C
The energy charge quotient has a value of unity (or, 1.00) when only ATP is present and a value of zero when only AMP is present.
The shipments of cattle were 84,205 head, with a value of $5,473,325.
In fact, it seems that all quantifiable properties obey the same relevant rule:
The Commodore PET was also released in 1977 with a price of $800.
The front of the train has a speed of 23 m/s.
The indefinite article is apparently at least allowable even when the relevant property refers to something previously mentioned, like in the second appearance of speed in the following example:
The train is moving with a speed of 5 mph. It then stops, and then slowly speeds up, until it is again moving with a speed of 5 mph.
(I don't have links to published versions of exactly these sentences, but I hope the following two examples come close enough: look for then accelerates back to a velocity of 25 m/s here, and for again with a speed of 250 r.p.m. here, the latter in the paragraph right below Fig. 2. See also Some analyses that (probably) don't work, B, below.)
The basic question is, why are we not using the definite article here? After all, in all these examples, the "of"-phrase would seem to be providing a further specification, a narrowing down, of the property in question. In other words, it would seem to be a standard prepositional phrase, which normally entails the definite article.
A fill-in-the-blank exercise
To help illustrate this last point, let's try an exercise. Fill in the blank ( __ ) in the following:
"The train is moving with a certain speed."
"What speed?"
"___ speed of 5 mph."
What did you put in the blank, "the," "a," or nothing? Surely, you put "the"?
I think I can show that the reason you put "the" is not that the speed is being mentioned for the second time—see Some analyses that (probably) don't work, A., below. The reason thus must be something else, and I think it is that "5 mph" provides a "narrowing down of possibilities to a single one," a "specifying," of the "speed."
And yet, just as surely, we would say,
The train is moving with a speed of 5 mph
even though here, too, it would seem that "5 mph" serves to "narrow down" the speed to a single value.
So, why the indefinite article? Why doesn't it matter that the numerical value narrows down the possibilities to just one? Why doesn't it matter if the relevant noun refers to something previously mentioned?
Categorical properties
The rule even seems to apply to the cases of categorical (as opposed to quantifiable) properties:
Forty-five percent of the population has a blood type of O.
He has a major in social work.
Wilde looked at the copy with an expression of surprise.
I am tempted to (semantically) analyze these examples as follows: they all involve a determinable property—i.e. a property that can get more specific—being more precisely characterized. In the case of quantifiable properties (inflation, speed, price, charge,…), we make them more specific by giving a numerical value. In the case of categorical properties, we restrict them by specifying the category: so a blood type can be A, B, AB or O; an expression can be one of surprise, fear, love, etc. And all of this talk of making things "more specific" would suggest the definite article, in contradiction to what is actually used.
Apparent counterexamples to the rule
Based on the preceding, we would expect the property of having an emotion (which can be more precisely characterized as happiness, sadness, boredom, surprise, anger, delight, …) should require the indefinite article as well. And yet, instead we have
God eternally has the emotion of compassion.
For example, consider what happens when the reader has the emotion of surprise.
And to make things really confusing, consider the property called having a property, which we further characterize by specifying the kind of property. Here there seems to be no pattern at all. You might hope that some of the usual rules would explain the particular choices of the articles in the sentences below (e.g. the property was—or was not—previously mentioned). However, if you look at the full texts, it will be apparent that such an explanation doesn't really work here:
The class of all spoons has the property of not being member of itself.
There is a thing which has a property of being the only writer of Waverly and of being Scotch.
As a set of points space has the property of containing points.
The construction of womanhood has a property of Otherness.
Here and then one encounters counterexamples to the paradigmatic cases discussed at the beginning. In the following, the first and the third boldfaced article are the counterexamples, while the second one is the (usual) indefinite article.
You may think that it is significant that we have mean speed. But, no…
Elite human athletes run 100-m races in about 10 s, at a mean speed of 10 m/s.
Ahead of the fleet lay a journey of some 420 miles, scheduled to last for thirty-five hours at a mean speed of twelve knots.
By now the Exeter and the Graf Spee were approaching each other at a mean speed of forty miles an hour.
Additionally, it seems that with the weight of used to be more acceptable in the past (see here), but for many decades already, with a weight of is preferred (see here).
The question summarized: All the examples above would seem to be clear cases of a prepositional phrase telling us "which one," i.e. "which value" of the many possible ones, and so we would expect the definite article in all of them. But instead, we usually use the indefinite article instead. Why? And why don't we use the indefinite articles in the case of having an emotion and in half the cases of having a property?
It could be that this an example where English hasn't yet really worked out what the rule should be. It could be that matters here are simply illogical (see here and here for some comments on why some usages of the English articles—including the zero article—may well be simply illogical). But are they illogical in the particular cases presented here? Or is there, after all, some rule, some pattern behind all of this?
Update 1 (with thanks to Edwin Ashworth): Sometimes when we more precisely characterize a determinable property, we do it by saying that it is the same as something else. In this case, we do use the definite article:
It has the color of deep rich caramel.
One copper coin has the value of a measure of rice.
I think the reason is this: when, to our ear, it seems that the "of"-phrase is coming from a transformation of a possessive phrase, the urge to use the definite article is simply too great. So we say It has the value of a Spanish milled dollar because it feels like it is a rephrasing of It has a Spanish milled dollar's value.
On the other hand, we don't have the same urge to say I have five grains of gold, with the value of $13, because we are not inclined to hear this as a rephrasing of a possessive phrase. And this for the simple reason that there simply is no corresponding possessive phrase. Note that with a $13's value doesn't sound right at all; it would instead have to be with a $13 value, which is not a possessive phrase. So the urge to use the definite article is not as great here, and, indeed, we rather have I have five grains of gold, with a value of $13, consistent with the previous examples. (Of course, a "decreased urge" to use the definite article is one thing; what is the actual reason why we use the indefinite article is another…)
Some analyses that (probably) don't work:
A. It has been suggested that the reason we put "the" in the fill-in-the-blank exercise (the second subsection of the text; see above) is that it is the second mention of the speed, i.e. because it is a reference to something previously mentioned. But if it were so, then this "the" would be obligatory, and it isn't:
"The train moving with a certain speed."
"What speed?"
"A certain speed. The point is, the speed is increasing."
Here the second-to-last sentence is used by the speaker to let his correspondent know that the precise value of the speed doesn't matter. (If it did matter, the speaker's reply would begin with the definite article, e.g. The speed of 5 mph.) But even so, in the sentence that follows, it is obligatory to say the speed and not a speed, because this really is a reference to something previously mentioned. Since in the second-to-last sentence the definite article is not obligatory, I conclude that when we do put it, we don't put it because it is referring to something previously mentioned (if that were the reason we put it, it would be obligatory). Instead, I think the reason is that "5 mph" provides a "narrowing down of possibilities to a single one."
B. It has been suggested that the reason we put the indefinite article in The train is moving with a speed of 5 mph is that this is the first reference to that speed. But this analysis does not work. Consider the following:
The train is moving with a speed of 5 mph. It then stops, and then slowly speeds up, until it is again moving with a speed of 5 mph.
Note that, at the end of the second sentence, it is again the indefinite article, even though we are referring to a previously mentioned speed (a point emphasized by the appearance of again). From this example, I conclude that the whole business of previous mention is irrelevant to what article is to be used here, although, of course, I still don't understand the grammatical reason why it is irrelevant, or why "5 mph" doesn't count as a "narrowing down of possibilities to a single one," like it (I think) does in the fill-in-the-blank exercise (see sub-subsection A, just above).
(I don't have links to published versions of exactly these sentences, but I hope the following two links come close enough: then accelerates back to a velocity of 25m/s, and again with a speed of 250 r.p.m., the latter in the paragraph right below Fig. 2.)
Best Answer
When the value is subject to change, then has a value of N is correct, it implies at different time or circumstances, the value may be different. Over time, the train will move at lots of different speeds. The value of gold changes depending on the market. The value read from a capacitor changes depending on charge time and voltage.
He has a major in social work. This implies he could have other majors in different subjects.
It has the color of rich deep caramel. This implies the color property does not change. Whatever "it" is, we wouldn't expect it to be blue tomorrow or black yesterday.
Using a or the implies variability in the value. There will be many examples, as you have cited above, that appear inconsistent with this. I'd put it down to most writers not being philosophers or logicians, and they write what sounds appropriate, using a or the almost interchangeably.