Heat capacity $C$ of an object is the proportionality constant between the heat $Q$ that the object absorbs or loses & the resulting temperature change $\delta T$ of the object. Entropy change is the amount of energy dispersed reversibly at a specific temperature. But they have the same unit joule/kelvin like work & energy. My conscience is saying these two are different as one concerns with temperature change and other only at a specific temperature. I cannot figure out any differences. What are the differences between heat capacity and entropy?
[Physics] Difference between heat capacity and entropy
entropythermodynamics
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There are two definitions of entropy, which physicists believe to be the same (modulo the dimensional Boltzman scaling constant) and a postulate of their sameness has so far yielded agreement between what is theoretically foretold and what is experimentally observed. There are theoretical grounds, namely most of the subject of statistical mechanics, for our believing them to be the same, but ultimately their sameness is an experimental observation.
(Boltzmann / Shannon): Given a thermodynamic system with a known macrostate, the entropy is the size of the document, in bits, you would need to write down to specify the system's full quantum state. Otherwise put, it is proportional to the logarithm of the number of full quantum states that could prevail and be consistent with the observed macrostate. Yet another version: it is the (negative) conditional Shannon entropy (information content) of the maximum likelihood probability distribution of the system's microstate conditioned on the knowledge of the prevailing macrostate;
(Clausius / Carnot): Let a quantity $\delta Q$ of heat be input to a system at temperature $T$. Then the system's entropy change is $\frac{\delta Q}{T}$. This definition requires background, not the least what we mean by temperature; the well-definedness of entropy (i.e. that it is a function of state alone so that changes are independent of path between endpoint states) follows from the definition of temperature, which is made meaningful by the following steps in reasoning: (see my answer here for details). (1) Carnot's theorem shows that all reversible heat engines working between the same two hot and cold reservoirs must work at the same efficiency, for an assertion otherwise leads to a contradiction of the postulate that heat cannot flow spontaneously from the cold to the hot reservoir. (2) Given this universality of reversible engines, we have a way to compare reservoirs: we take a "standard reservoir" and call its temperature unity, by definition. If we have a hotter reservoir, such that a reversible heat engine operating between the two yields $T$ units if work for every 1 unit of heat it dumps to the standard reservoir, then we call its temperature $T$. If we have a colder reservoir and do the same (using the standard as the hot reservoir) and find that the engine yields $T$ units of work for every 1 dumped, we call its temperature $T^{-1}$. It follows from these definitions alone that the quantity $\frac{\delta Q}{T}$ is an exact differential because $\int_a^b \frac{d\,Q}{T}$ between positions $a$ and $b$ in phase space must be independent of path (otherwise one can violate the second law). So we have this new function of state "entropy" definied to increase by the exact differential $\mathrm{d} S = \delta Q / T$ when the a system reversibly absorbs heat $\delta Q$.
As stated at the outset, it is an experimental observation that these two definitions are the same; we do need a dimensional scaling constant to apply to the quantity in definition 2 to make the two match, because the quantity in definition 2 depends on what reservoir we take to be the "standard". This scaling constant is the Boltzmann constant $k$.
When people postulate that heat flows and allowable system evolutions are governed by probabilistic mechanisms and that a system's evolution is its maximum likelihood one, i.e. when one studies statistical mechanics, the equations of classical thermodynamics are reproduced with the right interpretation of statistical parameters in terms of thermodynamic state variables. For instance, by a simple maximum likelihood argument, justified by the issues discussed in my post here one can demonstrate that an ensemble of particles with allowed energy states $E_i$ of degeneracy $g_i$ at equilibrium (maximum likelihood distribution) has the probability distribution $p_i = \mathcal{Z}^{-1}\, g_i\,\exp(-\beta\,E_i)$ where $\mathcal{Z} = \sum\limits_j g_j\,\exp(-\beta\,E_j)$, where $\beta$ is a Lagrange multiplier. The Shannon entropy of this distribution is then:
$$S = \frac{1}{\mathcal{Z}(\beta)}\,\sum\limits_i \left((\log\mathcal{Z}(\beta) + \beta\,E_i-\log g_i )\,g_i\,\exp(-\beta\,E_i)\right)\tag{1}$$
with heat energy per particle:
$$Q = \frac{1}{\mathcal{Z}(\beta)}\,\sum\limits_i \left(E_i\,g_i\,\exp(-\beta\,E_i)\right)\tag{2}$$
and:
$$\mathcal{Z}(\beta) = \sum\limits_j g_j\,\exp(-\beta\,E_j)\tag{3}$$
Now add a quantity of heat to the system so that the heat per particle rises by $\mathrm{d}Q$ and let the system settle to equilibrium again; from (2) and (3) solve for the change $\mathrm{d}\beta$ in $\beta$ needed to do this and substitute into (1) to find the entropy change arising from this heat addition. It is found that:
$$\mathrm{d} S = \beta\,\mathrm{d} Q\tag{4}$$
and so we match the two definitions of entropy if we postulate that the temperature is given by $T = \beta^{-1}$ (modulo the Boltzmann constant).
Lastly, it is good to note that there is still considerable room for ambiguity in definition 1 above aside from simple cases, e.g. an ensemble of quantum harmonic oscillators, where the quantum states are manifestly discrete and easy to calculate. Often we are forced to continuum approximations, and one then has freedom to define the coarse gaining size, i.e. the size of the discretizing volume in continuous phase space that distinguishes truly different microstates, or one must be content to deal with only relative entropies in truly continuous probability distribution models Therefore, in statistical mechanical analyses one looks for results that are weakly dependent on the exact coarse graining volume used.
For a reversible addition of heat, the entropy change is $\int \frac{dQ} {T}$, in other words the area under the graph of $\frac{1}{T}$ against $Q$ (heat added to system).
And yes, when a small amount of heat $\Delta Q$ is added, temperature T changes only a little, so $\frac{\Delta Q} {T}$ is well-defined. When added up this gives the integral.
Best Answer
https://en.wikipedia.org/wiki/Standard_molar_entropy?wprov=sfti1
$$dQ = T \ dS \tag1$$ $$dQ = C \ dT \tag2$$
Interesting, right? In $(1)$, the whole $T$ multiplies the infinitesimal $\frac{\text{J}}{\text{K}}$. In $(2)$ it's the opposite: the whole $\frac{\text{J}}{\text{K}}$ multiplies the infinitesimal $T$.
But you hinted that you knew that yourself already. Let's cut to the chase: both are different beasts entirely, just like heat and torque are not related just because they carry the same unit (joules are newton-meters, right?).
However, if you still want a defining difference between them, other than "they're just different", I'd give you this:
Entropy by itself is not useful and cannot even be measured. What is useful are changes in entropy, or how it differs from one state to the other. In this sense, it's akin to internal energy and enthalpy, for which only relative values matter. Heat capacity, on the other hand, can have its absolute value determined experimentally, and it won't depend on a reference value like entropy does. Its absolute value is immediately useful, if you will. In this sense, it's akin to pressure and specific volume, for which absolute values make sense.