Statistical Mechanics-Limit in canonical ensemble

In summary, the conversation is discussing the use of the approximation ##e^x \approx 1 + x## and when it is valid. This approximation is only valid for small values of ##x##. In the context of the conversation, the question is whether or not to expand the exponential term in a particular equation when the temperature is low. The answer is that for low temperatures, the exponential term becomes very large and therefore the approximation is not valid. As an approximation, only the first two terms should be considered in the equation.
  • #1
binbagsss
1,254
11

Homework Statement



question attached.

rotsm.jpg
My question is just about the size of the limit, how do you know whether to expand out the exponential or not (parts 2) and 4))

Homework Equations



for small ##x## we can expand out ##e^{x} ## via taylor series.

The Attempt at a Solution



Solutions given here for parts 2) and 4) respectively. (not full sol but only as relevant as needed for the question):
rot2sol.jpg
rot4sol.jpg


For parts 2) and 4) below I would have used the same approximation ##1+3e^{x}##, but would have expanded out the exponential as ##1+x##.

I'm just wondering how you know whether or not to expand this out or not, going by 'low ##T## many thanks.
 

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  • #2
binbagsss said:
For parts 2) and 4) below I would have used the same approximation ##1+3e^{x}##, but would have expanded out the exponential as ##1+x##.

I'm just wondering how you know whether or not to expand this out or not, going by 'low ##T## many thanks.
The approximation ##e^x \approx 1 + x## is only valid if ##x## is "small" (i.e., ##|x| \ll 1##).

For low ##T##, is the argument of the exponential small in any of the terms of ##Z## (other than the ##j = 0## term)?
 
  • #3
TSny said:
The approximation ##e^x \approx 1 + x## is only valid if ##x## is "small" (i.e., ##|x| \ll 1##).

For low ##T##, is the argument of the exponential small in any of the terms of ##Z## (other than the ##j = 0## term)?

would you not have to know an approx magnitude of the moment of inertia to know this? or should we be looking at planks constant, and Boltzmann constant, and physically feasible magnitudes of the moment of inertia, i hae no idea what these would be , and know they arent comparable so x is not small?
 
  • #4
The exponential factors in ##Z## have the form ##e^{-\beta E_j}##, where ##\beta E_j = \frac{\hbar^2 j(j+1)}{2 I k T}##.
As ##T## →0, ##\beta E_j## gets large (for any ##j > 0##). So, ##e^{-\beta E_j}## → 0 for any for any ##j > 0##. The approximation ##e^x \approx 1+x## doesn't apply here. The larger the value of ##j##, the faster ##e^{-\beta E_j}## → 0. So, as an approximation, you can keep just the ##j = 0## and ##j = 1## terms in ##Z##.
 

1. What is the canonical ensemble in statistical mechanics?

The canonical ensemble is a statistical mechanical model used to describe the behavior of a system in thermal equilibrium with a heat reservoir at a fixed temperature and volume. It is also known as the NVT ensemble, where N is the number of particles in the system, V is the volume, and T is the temperature.

2. What is the limit in the canonical ensemble?

The limit in the canonical ensemble refers to the limit as the number of particles in the system approaches infinity. This allows for the application of thermodynamic laws to the system, making it a useful tool in studying the behavior of large systems.

3. How is the canonical ensemble different from other ensembles?

The canonical ensemble differs from other ensembles in that it specifies the temperature of the system, whereas other ensembles may specify other variables such as energy or pressure. It is also used for systems in thermal equilibrium, while other ensembles may be used for non-equilibrium or time-varying systems.

4. What is the significance of the limit in the canonical ensemble?

The limit in the canonical ensemble is significant because it allows for the application of statistical mechanics to large systems. This makes it a powerful tool for studying the behavior of systems with a large number of particles, such as gases, solids, and liquids.

5. How is the canonical ensemble used in statistical mechanics?

The canonical ensemble is used in statistical mechanics to calculate the properties of a system at thermal equilibrium, such as the average energy, heat capacity, and entropy. It also allows for the calculation of thermodynamic quantities, such as the Helmholtz free energy and partition function, which can provide insights into the behavior of the system.

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