--%>

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 : Acid Solutions Choose the right answer

    Choose the right answer from following. Volume of water needed to mix with 10 ml 10N NHO3 to get 0.1 N HNO3: (a) 1000 ml (b) 990 ml (c) 1010 ml (d) 10 ml

  • Q : Net charge of a non-ionized atom

    Describe the net charge of a non-ionized atom?

  • 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 : Relationship between Pressure and

    The pressure-temperature relation for solid-vapor or liquid vapor equilibrium is expressed by the Clausis-Clapeyron equation.We now obtain an expression for the pressure-temperature dependence of the state of equilibrium between two phases. To be specific,

  • Q : Molarity of Barium hydroxide 25 ml of a

    25 ml of a solution of barium hydroxide on titration with 0.1 molar solution of the hydrochloric acid provide a litre value of 35 ml. The molarity of barium hydroxide solution will be: (i) 0.07 (ii) 0.14 (iii) 0.28 (iv) 0.35

  • Q : Colligative properties give atleast two

    give atleast two application of following colligative properties

  • Q : Carnot cycle show how a mathematical

    show how a mathematical definition of entropy can be obtauined from a consideration of carnot cycle?

  • Q : Explain Phase Rule The relation between

    The relation between the number of phases, components and the degrees of freedom is known as the phase rule. One constituent systems: the identification of an area on a P-versus-T with one phase of a component system illustrates the two degrees of freedom that

  • Q : Application of colligative properties

    Choose the right answer from following. Colligative properties are used for the determination of: (a) Molar Mass (b) Equivalent weight (c) Arrangement of molecules (d) Melting point and boiling point (d) Both (a) and (b)  

  • Q : Degree of dissociation The degree of

    The degree of dissociation of Ca(No3)2 in a dilute aqueous solution containing 14g of the salt per 200g of water 100oc is 70 percent. If the vapor pressure of water at 100oc is 760 cm. Calculate the vapor pr