Helmholtz Free Energy

In summary, we can express the free energy, energy, and entropy of a system with two states (one with energy 0 and one with energy \epsilon_0) as functions of temperature, T. We can also plot the average energy and entropy as a function of \tau = k_B T.
  • #1
stunner5000pt
1,461
2
[itex] \tau = k_{B} T [/itex]
a) Find the expression for the free energy as a function of the temperature of the system with two states - one iwth eneryg zero and one with energy [itex] \epsilon_{0} [/itex]

b) From the free energy find the expressions for the energy and entropy of the system

c) Plot the average energy and the entropy as a function of tau. [itex] \tau = k_{B} T [/itex]


Ok for a) wek now that
[tex] F = U - T \sigma [/tex]

Partition fun ction [itex] Z = \sum_{s} \exp(-\epsilon_{s}/\tau) = 1 + \exp(-\epsilon_{0}/\tau) [/itex]

so then
[tex] U = \frac{\epsilon_{0} \exp(-\epsilon_{s}/\tau)}{1 + \epsilon_{0} \exp(-\epsilon_{s}/\tau)} [/tex]

but I am not quite sure how to proceed with the calculation of the entropy, sigma ...

for b)
for entropy use this
[tex] \sigma = \left(\frac{\partial F}{\partial \tau}\right)_{V} [/tex]

but not sure about how to find the nergy for hte system... is it simply the expression wh9ich doesn't involve tau??

I was thiking a bit more

isnt helmholtz free enryg given by simple
[tex] F- F(0) = -\tau \log Z [/tex]??
 
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  • #2


a) The free energy, F, as a function of temperature, T, can be expressed as:
F(T) = -k_B T \ln (1 + e^{-\epsilon_0/T})

b) From the free energy, we can find the expressions for energy and entropy as:
U(T) = \epsilon_0 \frac{e^{-\epsilon_0/T}}{1 + e^{-\epsilon_0/T}}
and
S(T) = -k_B \left[(1 + e^{-\epsilon_0/T}) \ln (1 + e^{-\epsilon_0/T}) - e^{-\epsilon_0/T} \ln e^{-\epsilon_0/T} \right]

c) We can plot the average energy and entropy as a function of \tau = k_B T as follows:

Average energy:
The average energy, U(T), can be plotted as a function of \tau by substituting \tau = k_B T in the expression for U(T):
U(\tau) = \epsilon_0 \frac{e^{-\epsilon_0/\tau}}{1 + e^{-\epsilon_0/\tau}}

Entropy:
The entropy, S(T), can be plotted as a function of \tau by substituting \tau = k_B T in the expression for S(T):
S(\tau) = -k_B \left[(1 + e^{-\epsilon_0/\tau}) \ln (1 + e^{-\epsilon_0/\tau}) - e^{-\epsilon_0/\tau} \ln e^{-\epsilon_0/\tau} \right]
 

What is Helmholtz Free Energy?

Helmholtz Free Energy is a thermodynamic potential that describes the amount of energy that is available to do useful work in a thermodynamic system at a constant temperature and volume.

How is Helmholtz Free Energy related to other thermodynamic potentials?

Helmholtz Free Energy is related to other thermodynamic potentials, such as internal energy and enthalpy, through mathematical equations that depend on the specific conditions and constraints of the system being studied.

What is the significance of Helmholtz Free Energy in thermodynamics?

Helmholtz Free Energy is a useful concept in thermodynamics because it allows scientists and engineers to understand and predict the behavior of a system under different conditions, such as changes in temperature and volume.

How is Helmholtz Free Energy calculated?

Helmholtz Free Energy can be calculated using the equation F = U - TS, where F is the Helmholtz Free Energy, U is the internal energy, T is the temperature, and S is the entropy of the system.

What are some practical applications of Helmholtz Free Energy?

Helmholtz Free Energy has many practical applications, such as in the study of chemical and physical processes, the design of thermodynamic systems, and the development of renewable energy technologies.

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