This article discusses partition function of monatomic ideal gas which is given in Statistical Physisc at Physics Department, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia. Students in general are not familiar with partition function. This unfamiliarness was detected at a problem of partition function which was re-given in an examination in other dimensions that had been previously given in the lecture. Based on this observation, the need of a simple but comprehensive article about partition function in one-, two-, and three-dimensions is a must. For simplicity, a monatomic ideal gas is chosen.
Partition function for monatomic ideal gas is commonly discussed for three-dimensional case [1], but it is also interesting, in analogy and mathematical point of view, to discuss it in one-or two-dimension. Partition function can be viewed as volume in n-space occupied by a canonical ensemble [2], where in our case the canonical ensemble is the monatomic ideal gas system.
In order to understand this work reader must already familiar with Γ -integral and its relation with factorial ! n [3].
In general, a system of particles which obeys Maxwell-Boltzmann statistics, has a definition for partition function as: (2)
Energy of each particle, using the principle of quantum mechanics for single particle in a box, is given by [4]
For 3-D case as illustrated in Figure 1(a), it can be written that arXiv:1106.1273v2 [physics.pop-ph] 19 Aug 2011
then number of states of particles which have quantum number between j n and j j n n ∆ + or have energy between j ε and
Substitution Equation (5) into Equation (2) will give:
Figure 1. n-space for case of: (a) one-, (b) two-, and (c) three-dimensional monatomic ideal gas.
By using Equation (3) into Equation (6) and write the result in continuous form
The Γ -integral and its relation will give immediately the result of Equation (7) in a form of
Then, the next is 2-D case as illustrated in Figure 1(b). This case will give:
then number of state of particles which have quantum number between j n and j j n n ∆ + or have energy between j ε and
Substitution Equation (10) into Equation ( 2) will give
Following previous steps for 3-D case, but by using
, we can arrive at:
Finally by following the similar steps we can obtain that for 1-D case as illustrated in Figure 1(c)
then number of state of particles which have quantum number between j n and j j n n ∆ + or have energy between j ε and
Substitution Equation (15) into Equation (2) will give
Then following similar steps for 3-and 2-D case but by using 2
Equation ( 7), (12), and (17) can be solved using the following relations
and
It can be seen from Equation (8), (13), and (18) that there is a regularity in writing the partition function of monatomic ideal gas for 1-, 2-, and 3-D case as shown in Table 1.
Table 1. Comparison of partition function of monatomic ideal gas for 1-, 2-, and 3-D case.
Partition function
We can then say that the partition function of monatomic ideal gas can be written in general form, which is
where D , which is the dimension, can have value of 1, 2, or 3. Then thermodynamics property that similar to pressure p in 3-D case can also be defined in 2-and 1-D case. This property usually derived from Helmholtz free energy F , which is related to partition function through
and then
Table 2. Comparasion of pressure-like thermodynamics properties for monatomic ideal gas for 1-, 2-, and 3-D case.
Pressure-like properties Unit (SI)
We can then write a bit more generality the equation in Table 2 as follows: is the well-known outward-pointing pressure of the gas, exerted at its two dimensional boundary. Table 3. Equation of state for monatomic ideal gas for 1-, 2-, and 3-D case.
Equation of state
Table 3 shows us the equation of state of monatomic ideal gas for 1-, 2-, and 3-D case. The 3-D case is the most familiar form for the students, while this form is already taught since at senior high school.
Problems usually arise when the students do not understand where Equation ( 4), (9), and ( 14) are originated from and also why Equation ( 3) is needed. In this case the role of lecture is very important to guide them in the lecture.
It can be illustrated for 3-D case that Equation ( 4) is actually 8 1 volume of a sphere with radius j n and Equation ( 5) is 8 1 volume of a shell of the sphere with radius j n and thickness j n ∆ . Figure 1(a) shows the illustration. The number 8 1 appears since we consider only positive value of x n , y n , and z n , which lies only in one octane or 8 1 of total volume of the sphere, as this approach of explanation is suggested [1].
Following the approach for 2-D case as illustrated in Figure 1 So, perhaps it is also necessary to give an common picture about relation of j n with x n , y n , and z n in case of 1-, 2-, and 3-D case and also the factor 1, 4 1 , and 8 1 . Table 4 gives the illustration.
Table 4. Expression of j n and the factor in front of Equation ( 5), (9), and (14) for 1-, 2-, and 3-D case respectively.
The partition function of monatomic ideal gas system for 1-, 2-, and 3-D case has been reviewed and compared. General formulation has also been shown. Brief and simple explanation how theses partition functions derived is also given in order to help the students to understand it with olny few efforts.
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