Theory of three dimensional motion

Partition function; that the translational energy of 1 mol of molecules is 3/2 RT will come as no surprise. But the calculation of this result further illustrates the use of quantized states and the partition function to obtain macroscopic properties. The partition function is:

 
qtrans = Σ exp [- (n2x + n2y + n2z) h2/ (8ma2)/kT]  

= Σ exp [- n2x h2/ (8ma2)/kT] Σ exp [- n2y h2/ (8ma2)/kT] × Σexp [- n2z h2/ (8ma2)/kT]

= Σ exp [-n2x h2/(8ma2)/kT] Σ exp [-n2y h2/(8ma2)/kT] × Σexp [-n2z h2/(8ma2)/kT]

= qx qy qz

Each of the three partition function terms is like the one-dimensional term. We therefore can use:

qx = qy = qz = √∏/2 [kT/h2/(8ma2)] ½ 

to obtain, with V = a3,

qtrans = qx qy qz = (2∏mkT/h2)3/2 V

The Three dimensional translation energy: the three dimensional translation energy is derivative with respect to temperature can be used to reach an expression for the normal energy of three dimensional translational motions. Although qtrans depends on the particles and the volume of the container, the thermal energy (U - U0)trans has, for 1 mol of any gas in any volume the value 3/2 RT.

Distribution over quantum states: the distribution expressions for three dimensional motions can be derived by following the same procedure as we do for one dimensional motion before. First, however, we see that we can use one "effective" quantum number n in place of the three dimensional quantum numbers are nx, ny, and nz.

It is enough for us to deal with a quantity that shows the sum of the square of the equation of quantum numbers rather than with the individual values. We introduce the variable n defined by n2 = n2x + n2y + n2z.

Then the allowed energies are given instead of the more detailed manner than the previous one which we have done above. In using the effective quantum number n, we must recognize that there are number of states all with the same value of the energy. The display of states as point shows that, for large n, the additional number of states included when n increases by 1 is equal to 1/2πn2. Thus, if we use n as an effective quantum number, we must use gn = 1/2πn2.

Distribution over Quantum states: the distribution expressions for dimensional motion can be derived by following the same procedure as we did for one dimensional motion. First, however, we see that we can use one 'effective" quantum number n in place of the three quantum numbers nx, ny and nz.

(n2x + n2y + n2z) (h2/8ma2)

It is enough for us to deal with a quantity that shows the sum of the squares of the quantum numbers rather than with the individual values. We introduces the variable n defined by n2 = n2x + n2y + n2z. then the allowed energies are given by n2h2/(8ma2) instead of the more detailed, but no more useful, expression involving nx, ny and nz.

In using the effective quantum number n, we must recognize that there are a number of states all with the same value of n, or of energy εn. The number of states at this energy is the degeneracy gn. The display of states as points shows that, for large n, the additional number of states included when n increases by 1 is equal to ½ ∏n2. Thus if we use n as an effective quantum number we must use gn, ½ ∏n2 as the degeneracy.

   Related Questions in Chemistry

  • Q : Basicity order order of decreasing

    order of decreasing basicity of urea and its substituents

  • Q : Describe Point Groups. For any

    For any symmetric object there is a set of symmetry operations that, together, constitute a mathematical group, called a point group.

    It is clear from the examples that most molecules have several elements of symmetry. The H2O

  • Q : Product of HCl Zn Illustrate  the

    Illustrate  the product of HCl Zn?

  • Q : Latent heat of vaporization Normal

    Normal butane (C4H10) is stored as a compressed liquid at 90°C and 1400 kPa. In order to use the butane in a low-pressure gas-phase process, it is throttled to 150 kPa and passed through a vaporizer. The butane emerges from the vaporizer as a

  • Q : Define Virial Equation The constant of

    The constant of vander Waal's equation can be related to the coefficients of the virial equation. 

    Vander Waal's equation provides a good overall description of the real gas PVT behaviour. Now let us

  • Q : Molality of Sulfuric acid Choose the

    Choose the right answer from following. The molality of 90% H2SO4 solution is: [density=1.8 gm/ml]  (a)1.8 (b) 48.4 (c) 9.18 (d) 94.6

  • Q : Volume of solution containing solute

    What volume of solution contains 0.1 mole of the solute: (a) 100ml (b) 125ml  (c) 500ml (d) 62.5ml

    Choose the right answer from above.

  • Q : Calculating total vapour pressure

    Select the right answer of the question. The vapour pressure of two liquids P and Q are 80 and 600 torr, respectively. The total vapour pressure of solution obtained by mixing 3 mole of P and 2 mole of Q would be: (a) 140 torr (b) 20 torr (c) 68 torr (d) 72 torr

  • Q : Solubility product On passing H 2 S gas

    On passing H2S gas through a particular solution of Cu+ and Zn+2 ions, first CuS is precipitated because : (a)Solubility product of CuS is equal to the ionic product of ZnS (b) Solubility product of CuS is equal to the solubility product

  • Q : Colligative properties give atleast two

    give atleast two application of following colligative properties

2015 ©TutorsGlobe All rights reserved. TutorsGlobe Rated 4.8/5 based on 34139 reviews.