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Why isn't the free expansion of a gas in an adiabatic container isentropic?

Another way is see the difference is to look at the problem in terms microscopics. When you look close enough, you can imagine your gas as a number of particles distributed over some set of energy ...
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2 votes

Why isn't the free expansion of a gas in an adiabatic container isentropic?

You can also reason from the formula for the entropy change of an ideal gas. \begin{align} \Delta S &= C_V \ln\left(\frac{T}{T_0}\right) + Nk \ln\left(\frac{V}{V_0}\right), \end{align} where $...
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2 votes

Why isn't the free expansion of a gas in an adiabatic container isentropic?

There are only two mechanisms by which the entropy of a closed system can change: By heat transfer between the surroundings and the system at the location and temperature of the boundary between the ...
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7 votes

Why isn't the free expansion of a gas in an adiabatic container isentropic?

If you expand a gas adiabatically using a piston, the process is isoentropic. An adiabatic process is not isentropic unless it is also reversible. To be reversible, it must be carried out quasi ...
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12 votes

Why isn't the free expansion of a gas in an adiabatic container isentropic?

The free expansion isn’t reversible because the gas flows down a pressure gradient (that arises when you remove the piston). Any energy flow down a gradient generates entropy. In contrast, during the (...
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How does the adiabatic coefficient $γ$ vary with temperature (200K-20000K)?

The adiabatic coefficient is the ration of heat capacities at constant pressure and constant volume $\gamma = \frac{C_p }{ C_V} \overset{ideal gas}{\approx}=\frac{C_V + R }{ C_V}$. For the model air ...
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1 vote

How does the adiabatic coefficient $γ$ vary with temperature (200K-20000K)?

One place to start is the NASA Technical Reports Server (ntrs. nasa. gov - take out the spaces) and get Thermodynamic and Transport Properties of Air and the Combustion Products of Natural Gas and ...
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Thermodynamic adiabatic process, question regarding mathematical operations

Generally speaking you are right, there is no much meaning in integrating the different sides of an equation over different independent variables. But in the particular case that you have described, ...
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1 vote

Thermodynamic adiabatic process, question regarding mathematical operations

Since your question asks for mathematical justification I will answer purely mathematically, without digressing towards exact and inexact differentials or the physics behind your equations. Your first ...
-2 votes

Thermodynamic adiabatic process, question regarding mathematical operations

Before answering, some mistakes and some doubts about your question: mistake: first law of thermodynamics says that $dE = \delta W$, evaluating variation of internal energy as the work done on the ...
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1 vote

Proof of Caratheodory's theorem

Write the conservation of energy in differential form as $\delta Q = dU - \delta W$ and $\delta W = \sum_k y_kdx_k$, where $U$ is the internal energy and $x_k$ are the extensive parameters forming the ...
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1 vote

Problem related to adiabatic invariants

This question is a bit confusing since the amplitude does not depend on the oscillations, it's just the length of the pendulum. So either you calculate the rate of change $\frac{dA}{dt}$ or you're ...
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5 votes
Accepted

Understanding the use of $d$ and $\partial$ in thermodynamics

The partial and total derivatives are different things, but they are related via the Chain Rule: For $f(x,y,z)$, the differential of $f$ is: $$df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\...
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2 votes

Why is the value of the heat capacity ratio $\gamma$ never less than 1?

$$\gamma=\frac{C_p}{C_v}=\frac{C_v+R}{C_v}=1+\frac{R}{C_v}$$We know that R is a positive constant and the heat capacity at constant volume is greater than zero (a gas internal energy increases with ...
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2 votes

Why is the value of the heat capacity ratio $\gamma$ never less than 1?

Constant pressure heat capacity is always greater or equal to the constant volume heat capacity for perfect and real gases and for liquids, i.e., for every simple system described by $N$, $V$, $T$ ...
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3 votes

Why is the value of the heat capacity ratio $\gamma$ never less than 1?

A simple argument would be that the heat capacity ratio for an ideal gas is related to the degrees of freedom, $f$, via, $$\gamma=1+\frac{2}{f}\equiv\frac{f+2}{f}$$ Since the degree of freedom ...
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0 votes
Accepted

Why is the value of the heat capacity ratio $\gamma$ never less than 1?

Presumably by $\gamma$ you are referring the to the ratio of the specific heat at constant pressure $c_{p}$ to the specific heat at constant volume $c_V$ as applied to an isentropic process for an ...
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4 votes

Why is the value of the heat capacity ratio $\gamma$ never less than 1?

During constant volume heating, the gas's volume is fixed, and hence it does no work on its surroundings. During constant pressure heating, the gas is allowed to expand against its surroundings, ...
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2 votes

Question regarding using $dU=nC_vdT$

In thermodynamics, the correct equation to use for $C_v$ is in terms of the internal energy U rather than heat Q: $$C_v=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V$$In the case of an ...
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1 vote
Accepted

Question regarding using $dU=nC_vdT$

Yes you are right for the adiabatic process, the volume will definitely change, which may aptly make you question the validity of using $dU=nC_vdT$ Lets say in the given adiabatic process which you ...
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