How to calculate entropy from positions and velocities of gas molcules

In summary, calculating entropy from positions and velocities of gas molecules requires knowledge of the macrostate, which can be determined by defining the possible measurable macrostates and microstates. Entropy is a function of macroscopic variables such as volume, pressure, and temperature and cannot be accurately calculated without knowledge of these variables.
  • #1
olgerm
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How to calculate entropy from positions and velocities of gas molecules?

lets say we have 2 different gases. entropy should be bigger after mixing them, than before when these are separated. But how to calculate exact entropies by knowing only positions and velocities of gas molecules?
 
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  • #2
This is not defined.

Entropy is the log of the number of microstates for a given macrostate. You have defined the microstate but not the macrostate.
 
  • #3
Vanadium 50 said:
You have defined the microstate but not the macrostate.
isnt macrostate determined by the microstate? By knowing positions and velocities of gas molecules, I should know everything about the state of gas(including marcostate). Or you mean I should define possible(measurable) macrostates and microstates that correspond to every macrostate?
 
  • #4
olgerm said:
isnt macrostate determined by the microstate?

Of course not.

Suppose you had two otherwise identical particles, one moving to the left and one to the right at 1 m/s. Is your macrostate:
  • All states of two particles?
  • All states with an even number of particles?
  • All states with zero net momentum?
  • All two particle states with velocity differences of 2 m/s?
  • All two particle states with x-component of velocity differences of 2 m/s?
  • All states with p = 0 and E = m?
 
  • #5
Vanadium 50 said:
Is your macrostate:
I don't know.

Do you mean that I need relation between macrostates and mircrostates or macrostate of given microstate to calculate entropy?
 
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  • #6
olgerm said:
Do you mean that I need relation between macrostates and mircrostates or macrostate of given microstate to calculate entropy?

Vanadium 50 said:
Entropy is the log of the number of microstates for a given macrostate.

So yes, you need to know your macrostate. You have not defined the problem.
 
  • #7
In principle, if the microstate is "close enough to thermal equlibrium", with the particle density and kinetic energy density being constant down to very small scale, it could be possible to determine the pressure and temperature corresponding to that microstate and then calculate the entropy. But that's not how it's usually done and does not reflect the actual meaning of the concept of entropy. Entropy is a function of the macroscopic variables like volume, pressure and temperature, and can be defined without any knowledge about the atomic nature of matter.

If the microstate is such that there are rapid variations in density and the velocity distribution of the molecules isn't anywhere near the Boltzmann distribution, variables like entropy and temperature can't be defined in a meaningful way.
 
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1. How do you calculate entropy from positions and velocities of gas molecules?

The entropy of a gas can be calculated using the Boltzmann equation, which takes into account the positions and velocities of individual gas molecules. This equation is S = k ln(W), where S is the entropy, k is the Boltzmann constant, and W is the number of microstates (possible arrangements of molecules) that correspond to a given macrostate (observable properties such as temperature and pressure). By knowing the positions and velocities of gas molecules, we can determine the number of microstates and use the Boltzmann equation to calculate the entropy.

2. What is the relationship between entropy and the positions and velocities of gas molecules?

The positions and velocities of gas molecules directly affect the entropy of a gas. As the number of microstates increases (due to increased molecular movement), the entropy also increases. Conversely, when the number of microstates decreases (due to decreased molecular movement), the entropy decreases. Therefore, the positions and velocities of gas molecules play a crucial role in determining the entropy of a gas.

3. Can the positions and velocities of gas molecules affect the entropy of a gas without any change in temperature or pressure?

Yes, the positions and velocities of gas molecules can affect the entropy of a gas without any change in temperature or pressure. This is because the entropy of a gas is directly related to the number of microstates, which is determined by the positions and velocities of gas molecules. Even if the temperature and pressure remain constant, changes in the positions and velocities of gas molecules can lead to changes in the number of microstates and therefore, the entropy.

4. How do you determine the positions and velocities of gas molecules for calculating entropy?

The positions and velocities of gas molecules can be determined through experimental methods such as gas chromatography or molecular dynamics simulations. In gas chromatography, the positions and velocities of gas molecules can be indirectly inferred from the time it takes for the molecules to travel through a column. In molecular dynamics simulations, the positions and velocities of gas molecules can be directly calculated by solving the equations of motion for each molecule in a simulated environment.

5. What are the units of entropy when calculated from positions and velocities of gas molecules?

The units of entropy when calculated from positions and velocities of gas molecules are joules per kelvin (J/K). This is because the Boltzmann constant (k) has units of joules per kelvin, and the natural logarithm of the number of microstates (W) is a dimensionless quantity. Therefore, when multiplied together, the units of entropy become joules per kelvin.

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