Thermodynamics being a very broad topic having its applications in everyday activities, there are some equations that cannot be done away with due to its significance especially in the area of chemical engineering. These equations are known as the four Maxwell’s equations. When I was in the university, the derivation of these four equations were very difficult for my classmates and thus thought it wise to come up with a way to make things a bit easy and much more understandable for the school and the general science community in general. My way of deriving the four Maxwell’s equations begins with the first law of thermodynamics which is given by the formula,
∆U=∆q-∆w where U is the internal energy, q is the heat required and w is the work done. From the thermodynamics we know that w=PdV where P is the pressure required and dV is the change in the volume. Also we know that ∆q=TdS, where T is the absolute temperature and S is the entropy. Given that ∆U=0, we have TdS=PdV and then applying partial differentials to the result, we have, ∂T∂S=∂P∂V………………….. (A). Note very carefully and be watchful because this is where the actual trick and guidelines to the derivations begin; Looking at equations and dividing through, we have, (∂T¦∂V)=(∂P¦∂S)…. (1) Also due to the versatility of this derivation, the division could be anyhow and it you follow exactly the procedure, you’ll be able to arrive at the four Maxwell’s equations. All that i am trying to say is that you could have so many plausible starters but again due to the versatility of the derivation, any starter would work for the given guidelines if followed accordingly. The other plausible divisions are as follows, (∂S¦∂V)=(∂P¦∂T)….(2) (∂T¦∂P)=(∂V¦∂S)….(3)
(∂S¦∂P)=(∂V¦∂T)….(4). Any of these plausible could be used as starters. These four equations you see above are the four incomplete Maxwell’s equations. Another way of finding the four incomplete equations is just to use any of the four equations you see above as starters and follow some few guidelines and you will be just fine. So let’s say we choose equation (2) to start with. Here are the guidelines to follow strictly; Rule 1 states that, the following guidelines are to be applied on only the equation chosen and not the subsequent ones that are developed from the chosen one. The starting equation chosen is equation 2 above which is (∂S¦∂V)=(∂P¦∂T) Rule 2 states that switch the positions of the variables on one diagonal to get the second equation which is (∂T¦∂V) = (∂P¦∂S). Rule 3 states that switch the positions of the other variables on the other diagonal to give the third equation which is (∂S¦∂P)=(∂V¦∂T). Rule 4 states that switch the positions of the variables in each of the brackets giving as (∂V¦∂S)=(∂T¦∂P). Following the guidelines given above, we end up with the given equations; (∂S¦∂V)=(∂P¦∂T)…..(1) (∂T¦∂V) = (∂P¦∂S)…..(2) (∂S¦∂P)=(∂V¦∂T)…..(3) (∂V¦∂S)=(∂T¦∂P)…..(4). We can from the above equations above that comparing them to the ones we already had from the combination method are exactly the same. This is what we talk about the versatile derivation. The next time to do is to apply the subscripts and negatives to the equations where necessary. All these will come with some common guidelines so you just read carefully.¬¬¬
APPLICATION OF SUBSCRIPTS
The application of the subscripts is the easiest part of the derivation to grasp. Every one of Maxwell’s equations is made up of two brackets, each of which has a denominator and a numerator variable. Now the subscript for one bracket in any of the four equations is the denominator variable of the other bracket. For example,(∂S¦∂V)=(∂P¦∂T). For this particular equation, the subscript for the left hand bracket is the denominator variable of the right hand bracket which is T and the subscript for the left hand bracket is the denominator variable of the right hand side bracket. Applying this simple guideline, we come up with following equations but remember they are still incomplete unless we apply the negatives where necessary.
(∂S¦∂V)T=(∂P¦∂T)v (∂T¦∂V)S=(∂P¦∂S)v (∂S¦∂P)T=(∂V¦∂T)P (∂V¦∂S) P=(∂T¦∂P)s Now to the final part of the derivation which is the application of the negatives where necessary.
APPLICATION OF THE NEGATIVES.
The knowledge required here is the order of the alphabets, simple ha?. Yeah that’s all you need. What you need to do is to draw two diagonals on every one equation. Secondly, you fix arrow heads on the diagonals depending on the progression of the alphabets. Now the trick is that if you have the arrow heads pointing in the same direction, it means that both sides of that equation is positive but if the arrow heads are pointing in different directions, then we negate one side of the equation. Thus applying the following guidelines to the incomplete four Maxwell’s equations;
(∂S¦∂V)T = (∂P¦∂T)v Since they are pointing in same directions that is downwards then both sides of the equation are positive.
(∂T¦∂V)S = – (∂P¦∂S)v Since the arrows are pointing in different directions, that is one arrow head pointing upwards and the other pointing downwards this means that according to the rules above, we would have to negate one side of the equation.
(∂S¦∂P)T = – (∂V¦∂T)P Since they are pointing in different directions that is one the arrow heads pointing upwards and the other pointing downwards, it means that we have to negate one side of the equation.
(∂V¦∂S) P = (∂T¦∂P)s Since the arrows are pointing in the same direction that is upwards, it means both sides of the equation are positive. . This means that the final four of Maxwell’s equations are:
(∂T¦∂V) S= -(∂P¦∂S)v
This concludes the Addo-Yobo Alfred’s derivation of the four Maxwell’s equations.
Due to the format of submission, the arrows at the latter on how to derive the negatives are not clearly shown. But i guess from the description on how to derive the negatives, there would be no problems. Just draw diagonals, across the brackets, and shown the progression of the alphabets by placing an arrow to show the progression. After showing the progression of the alphabets by the arows, we place negative sign on one side of equation if the arrows are pointing to different directions and positive when they are pointing in the same directions.
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