Why Heat capacity and Entropy are different quantities when they have both the same units?
It means they are the same.
They have different dimensions. Cv for example has a dimension of energy/1K (1K - 1grad of absolute temperature scale). Entropy has a dimension of energy only. Because it is a whole energy (heat) spent on a sample to reach some temperature T minus work done due to sample expansion.
Cp (or Cv) and S both have units of kJ/kg/°C. We might use Cp or Cv in the calculation of S, for example of an ideal gas S2-S1=Cp*ln(T2/T1)-R*ln(P2/P1) which is the same as Cv*ln(T2/T1)+R*ln(V2/V1). But specific heat and entropy are two very different things. Specific heat is related to the energy content of a substance; whereas, entropy is related to the molecular structure and order. I recommend the greatest text ever written on the subject: Introduction to Classical Thermodynamics by Gordan van Wylen and Richard Sonntag.
Dear Dudley J Benton, you are absolutely right, pointing out the formal equality of the dimensions of entropy and heat capacity and their different content. I just wanted to emphasize that the energy spent on one degree of heating and on heating to given temperature still differ in meaning and can be a source of errors.
Indeed, not always, physically different quantities have got different physical dimensions, e. g. action and angular momentum. Is that a deficit in contemporary physics?!
Dear Peter Enders,
you know that action is not only the angular momentum, but also is a product of energy and time and a less known product of electron charge e and quantum of magnetic flux μ as in the formula h=eμ. Even the mass of elementary particles may be connected with something as yet unknown x to satisfy the formula h=mx. If so, we are able to describe many if not all the physical quantities in terms of action quanta.
Answering your question, It is not a deficit in contemporary physics but seems the action quantum to be a building block of everything in nature. Is not it?
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