Wikipedia says about the equation of state as:  For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form ${\displaystyle f(p,V,T)=0}$  But why are there only three variables in the equation of state? Why can't say entropy and other state variables be a part of the equation of state?
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A given thermodynamical system has a number of state variables, not all of which are independent. Suppose that we have a system which can be specified by $k+1$ extensive variables: $U,X_1,\cdots,X_k$. We also have $k$ intensive variables which are conjugate to the $k$ extensive variables $X_i$. To what extent are these $2k+1$ variables independent?   For example, we can always express the internal energy in terms of its natural variables: $$U=U(S,V,N_1,\cdots, N_{k-2})$$ But what about in terms of other variables? How about exchanging an extensive variable for its conjugate internal variable, say $T$ in place of $S$: $$U=U(T,V,N_1,\cdots,N_{k-2})?$$ Or perhaps replacing $T$ by some unrelated intensive variable, say $p$ in place of $S$: $$U=U(p,V,N_1,\cdots,N_{k-2})?$$ Will any combination of the remaining thermodynamic variables serve to functionally determine $U$? If not, then what exactly relates these $2k+1$ variables?
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I was wondering if temperature is related to the average translational kinetic energy of the molecules then why does the average kinetic energy of say a moving object not affect the temperature? What is the mechanism regarding these vibrational motions that imparts temperature. Also do a gas and liquid at the same temperature possess the same kinetic energy? and a heavier molecule traveling at the same speed as a lighter molecule will have a greater temperature?
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The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy.   Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero.  So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics?
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What is meant by ? My professor tells me "heat content". That literally makes no sense. Heat content, to me, means internal energy. But clearly, that is not what enthalpy is, considering: $H=U+PV$ (and either way, they would not have had two words mean the same thing). Then, I understand that $ΔH=Q_{p}$. This statement is a mathematical formulation of the statement: "At constant pressure, enthalpy change may be interpreted as heat." Other than this, I have no idea, what $H$ or $ΔH$ means.  So what does $H$ mean?
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What is latent heat of vaporization ($L_v$) in the first place?  seems to indicate that it is the energy used in overcoming intermolecular interactions, without taking into account at all any work done to push back the atmosphere to allow for an increase in volume when a liquid boils.  If that is so, then would it be correct that $L_v$ decreases as boiling point rises, because at the higher boiling point, less energy is required to overcome the weaker intermolecular interactions?  Otherwise, would it then be correct to say that $L_v$ increases as boiling point rises, assuming constant-volume?  Let's say there is a beaker containing 1 kg of liquid water at the boiling point, and an identical beaker, containing an identical amount of water at the same temperature. However, this second beaker is perfectly sealed such that the volume of its contents, both liquid and gaseous, will not change.  Since both liquids are at the boiling point, applying heat should cause boiling to occur. Compared to the first beaker, then, would the second beaker (in theory) require more or less heat for its 1 kg of liquid water to completely boil?  Thanks!
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Entropy is defined in my book as $\Delta\ S = \frac{Q}{T}$. To derive the formula it says that entropy should be directly proportional to the heat energy as with more energy the particles would be flying all over the place more rapidly. This makes sense. But then it says that entropy must be inversely proportional to the temperature at which it is added. To be precise  Heat added to a system at a lower temperature causes higher entropy increase than heat added to the same system at a higher temperature.  How does this make intuitive sense? EDIT 1: I found an answer . I think this makes sense. Can anyone read it and tell me if it's correct or not?  Rough intuitive answer: Adding heat to a system increases the system's total energy. This gives more kinetic energy to distribute among the particles in the system, increasing the size of the system's phase space and hence its entropy. However, since momentum is proportional to the square root of kinetic energy, it is clear that the total momentum phase space cannot grow exponentially as you add more kinetic energy; so the rate of growth of its logarithm must taper off. Since entropy is proportional to the log of the size of the phase space, we see that each additional unit of heat must increase the entropy by less and less. Now, for technical details. The reason why the relationship is precisely $dS = \delta q/T$ rather than some more complicated relationship that depends on the nature of the system is that temperature is defined such that this is true.  We know that during first-order phase transitions, the temperature actually stays constant as you add heat! This corresponds to a region where adding heat actually does cause an exponential increase in phase space. Finally, note that some systems actually have negative temperature, and adding more heat to them actually makes the temperature even more negative; so adding heat actually decreases the phase space. So the intuitive answer in the previous paragraph shouldn't be taken too seriously.  EDIT 2: Moreover,  answer explains why there needs to be an additional factor of temperature along with heat in the definition of entropy so as to be consistent with the second law.  This is why heat flows from hot objects to cold objects: the entropy change of the hot object is negative, while that of the cold object is positive. The magnitude of entropy change for the cold object is greater. $\Delta S = \frac{Q}{T_c}-\frac{Q}{T_h} &gt; 0$ as $T_h &gt; T_c$  Keep in mind that entropy increases with temperature. This can be understood intuitively in the classical picture, as you mention. However, at higher temperatures, a certain amount of heat added to the system causes a smaller change in entropy than the same amount of heat at a lower temperature. The formula is $\Delta S = \frac{Q}{T}$. The change in entropy is related to heat. Remember that heat is a form of energy transfer, not energy; we can talk about heat only when some change takes place. Please do point out if there is anything wrong with the answers above.
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Some physical quantities like position, velocity, momentum and force, have precise definition even on basic textbooks, however energy is a little confusing for me. My point here is: using our intuition we know what momentum should be and also we know that defining it as $p = mv$ is a good definition. Also, based on Newton's law we can intuit and define what forces are.  However, when it comes to  many textbooks become a little "circular". They first try to define work, and after some arguments they just give a formula $W = F\cdot r$ without motivating or giving intuition about this definition. Then they say that work is variation of energy and they never give a formal definition of energy. I've heard that "energy is a number that remains unchanged after any process that a system undergoes", however I think that this is not so good for three reasons: first because momentum is also conserved, so it fits this definition and it's not energy, second because recently I've heard that on general relativity there's a loss of some conservation laws and third because conservation of energy can be derived as consequence of other definitions.  So, how energy is defined formally in a way that fits both classical and modern physics without falling into circular arguments?
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I am studying entropy and its hard for me to catch up what exactly is entropy.  Many articles and books write that entropy is the measure of randomness or disorder of the system. They say when a gas system is let expand the randomness increases etc. But they end up saying $\frac{\mathrm dQ}{T}$ is the measure of increase in randomness and is called the entropy.  Even if I believe that entropy is the measure of randomness of the system I don't understand:   How does $\frac{\mathrm dQ}{T}$ hold the information about the randomness of the system? How is entropy independent property of any system. I suppose that any two parameter in the equation $PV=nRT$ should completely describe the system. Why would we need entropy?   Thank you.
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Is the entropy of every system zero at the absolute zero?  Or is it taken to be zero at the absolute zero?  Are there systems that doesn't reach zero entropy even till absolute zero?
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I was studying a book about thermodynamics of nanosystems and I got stuck with this question in my mind which I couldn't find an answer for.  For instance, does a single water molecule have a state, like being liquid or something? Or is one molecule of TiO2 solid or gas? I would appreciate if anyone could clarify this.  I should state that I am not considering a macromolecule in my case.
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I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector.  I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong.   But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$
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I'm having a discussion with someone. I said that it is -even theoretically- impossible to reach $0$ K, because that would imply that all molecules in the substance would stand perfectly still.  He said that this isn't true, because my theory violates energy-time uncertainty principle. He also told me to look up the Schrödinger equation and solve it for an oscillator approximating a molecule. See that it's lowest energy state is still non-zero.  Is he right in saying this and if so, can you explain me a bit better what he is talking about.
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I often heard that conservation of momentum is nothing else than Newton's third law.   Ok, If you have only two interacting particles in the universe, this seems to be quite obvious.   However if you have an isolated system of $n$ ($n &gt; 2$) interacting particles (no external forces). Then clearly Newton's third law implies conservation of total momentum of the system. However presuppose conservation of total momentum you only get:  $$ \sum_{i\neq j}^n \mathbf F_{ij} = \frac{d}{d t} \mathbf P = 0 $$  Where $\mathbf F_{ij}$ is the forced acted by the $i$th particle upon the $j$th particle and $\mathbf P$ is the total linear momentum.  But this doesn't imply that $\mathbf F_{ij} = -\mathbf F_{ji}$ for $j \neq i$.  So does conservation of momentum implies Newton's third law in general or doesn't it? Why?
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I am wondering about the partition function of the classical microcanonical ensemble. It contains Planck's constant and also an indistinguishability argument about the particles I am looking at and I find this confusing for the following reasons:  If I describe for example an ideal gas using classical mechanics, there is no reason why I should assume that there is a problem to distinguish the gas molecules or why $\hbar$ should occur there.   I mean, how could people like Gibbs etc. derive this equation if they did not know about QM?   In some sense, there has to be a reason why we have these two QM properties in there and maybe it is just to get the limit to QM right, but is this really the only reason for this?  I would love to receive a few ideas about this strange definition by this community, in order to get impressions, why we model a (NVE) ensemble this way?
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In the Landau theory of phase transitions, one typically considers a "free energy" $F$ as a function of the temperature $T$ and the "order parameter" $\psi$: $F(T, \psi)$.  For the sake of clarity, let's consider the liquid-gas transition, in which the order parameter is usually taken to be the difference of densities between the ordered phase and the disordered phase.  In this case, it looks that the order parameter $\psi$ is closely related to the the volume $V$, a global variable of state.  What is the correct interpretation of $\psi$ as a thermodynamic variable?   I consider several logical possibilities, but all interpretations look troublesome to me:    $\psi$ is a variable of state replacing the volume. In this case I have a few related concerns:   Is the "free energy" $F$ the Helmholtz or the Gibbs free energy?    If $F$ is the Gibbs free energy, $F$ should depend on the temperature $T$ and the pressure $p$ (the conjugate variable of the volume), so it shouldn't depend on $\psi$.  If it is the Helmholtz free energy, it's ok that it depends on $\psi$, but the experiments are usually carried out at constant pressure, not constant volume.  Usually, state variables are something one can select at will [for instance, think of the phase diagram $(p, T)$]. However, the order parameter $\psi$ is not selected at will, but it is selected by the system as a result of the minimization process according to the Landau theory.  $\psi$ is yet another variable of state in addition to volume and temperature.    Why the pressure is not usually taken into account as a variable of state, and only the temperature is considered? What is the conjugate variable to $\psi$? Can really $\psi$ and $V$ be taken as independent?  $\psi$ is not a variable of state, is something else.   What else can $\psi$ be from the thermodynamic point of view? Same question as above, why is pressure not usually taken into account in Landau theory?
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Is the degree of freedom defined in  same as the degree of freedom in ? If not what is the formal definition of degree of freedom in thermodynamics?
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Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each other are zero.  The way I understand it, these conditions correspond directly to us being able to do the Hamilton-Jacobi transformation, which is roughly equivalent to saying that the $2n-1$ conserved quantities are the initial conditions of the problem, which itself is a way of saying that the map from the phase space position at some time $t_0$ to that at time $t$ is invertible. But, if the last statement is right, why are there systems which are non-integrable at all? Shouldn't all systems' trajectories be uniquely determined by the equations of motion and initial conditions? Or is it that all non-integrable systems are those whose Lagrangians can't be written (non-holonomic constraints, friction etc)?  I've heard that Poincare proved that the gravitational three-body problem in two dimensions was non-integrable, but he showed that there were too few analytic conserved quantities. I don't know why exactly that means non-integrability, so if someone could help me there that would be great too.