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You and Lubos are integrating the expression $dU = -pdV$ for a constant composition system, but this expression is only valid for constant entropy $S$. During integration you maintain $T$ constant, bu …

Heat is not a property of a system. Heat is a process function. Temperature is a property of a system because is a state function. For instance, the state of a simple gas is given by temperature, pres …

Supercritical fluids are well described by real and ideal gas laws. A common equation of state for both liquids and solids is $$V_m = C_1 + C_2 T + C_3 T^2 - C_4 p - C_5 p T$$ where $V_m$ is molar …

The thermodynamic definition of temperature is $$T \equiv \left( \frac{\partial S}{\partial U}\right)^{-1} $$ where $S$ is the thermodynamic entropy of the system and $U$ its internal energy. The th …

The fundamental quantity in thermodynamics is entropy, which is a function of $n$-variables $S=S(x_1, x_2,...,x_n)$. … Using the definition of average $$\langle A \rangle = \int A P \mathrm{d}x_1 \mathrm{d}x_2 \cdots \mathrm{d}x_n$$ the demonstration of the central result of linear nonequilibrium thermodynamics $$\ …

There are different answers to your question. I will put here what I believe is the more popular in the literature. We start from the quantum mechanical expression for the energy average $$ \langle …

Taking $\mathcal{E}= k_\mathrm{B}T$ and pretending that Boltzmann constant is a "historical artifact" because we are not measuring temperatures in "units of energy" is like taking $E = mc^2$ and $E=\h …

Mechanical work $dW_\mathrm{mech} = -pdV$ is due to a volume change for a pressure $p$. But other kind of thermodynamic works exist: Chemical work $dW_\mathrm{chem} = \mu dN$ involves change in comp …

First, using $(P,T,V)$ at equilibrium is both redundant and insufficient. It is redundant because you are using two conjugate variables $P$ and $V$ and you only need one of them. It is insufficient, b …

The classical thermodynamics version $\Delta S \ge 0$ for isolated systems is not fundamental. First it doesn't apply to open systems and has to be replaced by $\Delta_i S \ge 0$. … Precisely that is the reason why thermodynamics was invented to deal with such observations and complement Newtonian mechanics. …

The instantaneous temperature of a gas system is a fluctuating quantity given by $$\tilde{T} \equiv \frac{2\tilde{K}}{k_\mathrm{B} N_\mathrm{df}}$$ here $ \tilde{K}$ is the fluctuating kinetic energy …

In statistical thermodynamics, when using the method of Lagrange multipliers, we obtain an expression as $$-\ln \rho = \alpha + \beta H$$ where $\alpha$ and $\beta$ are the multipliers to be determined …

In classical thermodynamics heat and work have to be represented by inexact differentials $\delta Q$ and $\delta W$, but once you extend the classical thermodynamics space of variables with the variable … A good discussion of why heat and work are exact differentials $dQ(t)$ and $dW(t)$ when you include time is given in the well-known textbook by Prigogine & Kondepudi: «Modern thermodynamics: from heat …

The internal energy of an ideal gas only depends on temperature. This result is not restricted to free expansions, but is completely general. One arrives to this conclusion by using the Helmholtz equa …

Heat is a process quantity, not a system quantity, and there is no indicator that you can attach to the object, but you could use a thermometer to record the distribution of temperatures of the object …