5
votes
Why is it so much harder to keep the same room cool than to keep it warm?
I am asking why the cool room warms up faster (loses the cold after you turn off the AC) then the warm room loosing cool down (lose the heat after you turn off the heater).
That's not a universal ...
- 33.7k
5
votes
The entropy given by stefan Boltzmann's law looks remarkably similar to the volume of the sphere; $S(T)=\frac{4}{3}\sigma T^3$
It's a coincidence, as the lack of $\pi$ indicates. The entropy per surface of a blackbody in $D$-dimensional space is $\frac{D+1}{D}\sigma T^D$. (You can deduce it e.g. by generalizing this.) By ...
- 21.7k
5
votes
Accepted
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
It's not unreasonable to be confused about this. Let's say you have a function $f$ of one variable and $f'$ is its derivative. Then we have
$$\frac{d}{dx} f(c-x) = \color{red}{-}f'(c-x)$$
via the ...
- 50.8k
3
votes
Accepted
Is there a clear analogy here between volume and entropy? Looking at: $ \frac{dQ}{T} = dS, \frac{dW}{P} = dV$
I’d prefer to write your equations as $$\int\frac{q_\text{rev}}{T}=\Delta S;$$$$\int\frac{w_\text{rev}}{P}=\Delta V;$$
where $q_\text{rev}$ and $w_\text{rev}$ correspond to infinitesimal reversible ...
- 13.6k
2
votes
Accepted
Mixing identical gases in different states (Gibbs paradox)
Yes, it works without any problems. I’ll just add that you can do it all in one go by modifying the expression of your entropy:
$$
S =nc_v\ln(T/T_0)+nR\ln\left(\frac{V/n}{ V_0/n_0}\right)+ns_0
$$
with ...
- 1,145
2
votes
Accepted
How to derive Shannon Entropy from Clausius Theorem?
You can't. It is not possible to derive a more general formula from a less general one. Of course, one can find hints for the generalization, but the validity of the generalization has to be proved ...
- 23.3k
2
votes
How to derive Shannon Entropy from Clausius Theorem?
These are not the same.
Shannon entropy (Information entropy), $H_\alpha=-\sum_i p_i\log_\alpha p_i$ applies to any system with specified probabilities $p_i$.
Boltzmann entropy, defined via the famous ...
- 37.2k
2
votes
"Universal" versus "nonuniversal" in the topological entanglement entropy
The idea is that $\gamma$ is constant throughout a phase - i.e., it can only change at a phase transition, i.e. when the gap closes. Thus, it can be used as a signature of the phase - in this sense, ...
- 16.7k
2
votes
To clarify the entropy/ energy/ Gibbs Eqs
Initial State:
$$T_1=313\ K$$$$V_1=V$$$$n_1=n$$$$P_1=\frac{n_1RT_1}{V_1}=\frac{nR(313)}{V}$$
Final State:$$T_2=313\ K$$$$V_2=V$$$$n_2=n$$$$P_2=\frac{n_2RT_2}{V_2}=\frac{nR(313)}{V}$$
The final state ...
- 27.7k
1
vote
How to derive Shannon Entropy from Clausius Theorem?
A better approach would be to use the Shannon Entropy to derive Gibbs entropy: $S=−k∗∑p_n∗ln(p_n)$. The two equations are very similar and therefore it is much easier understand. From there it is ...
- 79
1
vote
Is incompressible fluid flow isentropic?
No. For a viscous fluid, flow isn't isentropic. Viscosity dissipates kinetic energy as heat, generating entropy.
- 3,239
1
vote
What is the measure of decoherence?
In time domain decoherence is usually characterized by the rate of decay of coherences, i.e., the non-diagonal density matrix elements. This, of course, is dependent on the mechanism that causes ...
- 37.2k
1
vote
Is $\frac{\Delta Q}{\Delta S} = T $ a meaningful equation and what would it actually represent?
“$\Delta Q$” and “$dQ$” imply that there’s some function $Q$ that we can reasonably take the difference or derivative of. This isn’t the case; we can instead talk about $Q$, the heating, or $q$ (...
- 13.6k
1
vote
What stops entropy from forming if you extract 100% of heat flow and use it for work?
What exactly is happening here, why is entropy created and why does
that make it irreversible?
For the irreversible isothermal expansion less work is done because less heat is taken from the hot ...
- 55.5k
1
vote
Accepted
Finding equation of internal energy in adiabatic process
I think the unmentioned assumption here is that the adiabatic process in question is quasistatic/reversible. In this case no heat transfer means no entropy change, $dS=0$, that is
$$
dU=TdS - PdV=PdV
$...
- 37.2k
1
vote
Is there a clear analogy here between volume and entropy? Looking at: $ \frac{dQ}{T} = dS, \frac{dW}{P} = dV$
The general expression for the first law (using the OP sign convention) is: $dE = \delta q + \delta w$. When the process is reversible it is possible to write: $dE = TdS + PdV$. This expression is the ...
- 11.8k
1
vote
How to find a complete understanding the 2nd law of Thermodynamics in terms of forms?
The proper way to deal with Physics, particularly Thermodynamics, is to use Mathematics as a language to say something about the world and not to ask how to make the world fit a particular ...
- 23.3k
1
vote
Accepted
How to find a complete understanding the 2nd law of Thermodynamics in terms of forms?
For your first question, I guess you could, but it would be a mere translation of the inequalities of integrals. For example, when people write $dS\geq \delta Q/T$, they mean $\int dS (=\Delta S)\geq \...
- 1,145
1
vote
Accepted
What does "energetically favorable" mean?
There is a clear difference between energetically favourable reactions and spontaneous reactions.
Energetically favourable reactions are reactions with $\Delta H <0$ and spontaneous reactions are ...
1
vote
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
The point is that $E$ is fixed, so that if you raise $E_1$ by some amount $\delta x$ you have to lower $E_2$ by the exact same amount $\delta x$. This means that $E_1$ and $E_2$ can not be varied ...
- 7,006
1
vote
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
In my opinion the notation is very bad. $S_2$ is a function of a single variable (at least in this context here) and it simply does not make sense to compute partial derivatives with respect to two ...
- 3,482
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