Plot Heat Capacity vs Temperature for a 2 state system microcananonical ensemble

You can also plot the derivative of the left side, ##e^u##, to see where it crosses ##-1##. This will give you a rough idea of where the root lies.
  • #1
binbagsss
1,305
11

Homework Statement



I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

and need to sketch ##C## vs. ##T##

Homework Equations



See above

The Attempt at a Solution



I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

Considering asymptotic limits I have:

##C \to e^{-\frac{\epsilon}{K_{B}T}} ## as ##T \to 0##
##C \to \frac{1}{T^{2}} ## as ##T \to \infty##

The solution is attached.

So from these limits I get the shape at these ends, and deduce there is a maximum to allow me to sketch the rest of it.

I am unsure how to deduce this maximum?

Differentiating gives quite a mess and it seems that it should be obvious to conclude the maximum is at ## \epsilon / K_{B} ##, or at least a better method to find this point? (My knowledge of graph sketching is quite poor).

Many thanks in advance.
 

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  • #2
binbagsss said:

Homework Statement



I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

and need to sketch ##C## vs. ##T##

Homework Equations



See above

The Attempt at a Solution



I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

Considering asymptotic limits I have:

##C \to e^{-\frac{\epsilon}{K_{B}T}} ## as ##T \to 0##
##C \to \frac{1}{T^{2}} ## as ##T \to \infty##

The solution is attached.

So from these limits I get the shape at these ends, and deduce there is a maximum to allow me to sketch the rest of it.

I am unsure how to deduce this maximum?

Differentiating gives quite a mess and it seems that it should be obvious to conclude the maximum is at ## \epsilon / K_{B} ##, or at least a better method to find this point? (My knowledge of graph sketching is quite poor).

Many thanks in advance.
bump
 
  • #3
The only suggestion I'd have is to let ##u=\frac{\epsilon}{k_\text{B}T}## and find the extremum of
$$\frac{u^2 e^u}{(1+e^u)^2}.$$ It shouldn't be that messy.
 
  • #4
vela said:
The only suggestion I'd have is to let ##u=\frac{\epsilon}{k_\text{B}T}## and find the extremum of
$$\frac{u^2 e^u}{(1+e^u)^2}.$$ It shouldn't be that messy.

I have:

##2e^u+2+u-e^u u =0 ## , unsure of where to go now...
 
  • #5
You'd have to solve that numerically. To get a qualitative idea of where the root lies, you can rewrite that equation as
$$e^u = \frac{u+2}{u-2}.$$ Plot graphs of the two sides of the equations and see where they intersect.
 

FAQ: Plot Heat Capacity vs Temperature for a 2 state system microcananonical ensemble

What is a 2 state system microcananonical ensemble?

A 2 state system microcanonical ensemble is a thermodynamic ensemble that describes a system with only two possible states or configurations. This ensemble assumes that the system is isolated and the energy is constant. It is often used to study simple systems, such as ideal gases.

What is a plot of heat capacity vs temperature?

A plot of heat capacity vs temperature shows the relationship between the heat capacity of a system and its temperature. Heat capacity is a measure of the amount of heat energy required to raise the temperature of a system by one degree. This plot can provide insights into the thermal behavior and energy storage of a system.

How is heat capacity calculated in a microcanonical ensemble?

In a microcanonical ensemble, heat capacity is calculated by taking the derivative of the system's energy with respect to temperature. This is known as the microcanonical heat capacity and it provides information about the fluctuations in energy as the temperature changes.

What does a flat region in the heat capacity vs temperature plot indicate?

A flat region in the heat capacity vs temperature plot indicates a phase transition or a change in the thermodynamic behavior of the system. This can occur when there is a sudden change in the number of accessible states or configurations at a certain temperature.

How does the number of particles affect the plot of heat capacity vs temperature?

The number of particles in a system can affect the shape of the heat capacity vs temperature plot. For a fixed energy, as the number of particles increases, the heat capacity becomes more peaked and the transition region becomes narrower. This is known as the equipartition effect and it is a result of the increased number of degrees of freedom in the system.

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