--%>
Assignment Help - homework help

Theory of one dimensional motion

For motion in one dimension, the distribution of the molecules over quantum states, speeds, and energies can be deduced.

Here we show that the energy of a macroscopic gas sample can be described on the basis of our knowledge of the quantum states allowed to the molecules of the gas and the distribution expressed by the Boltzmann expression. We begin by studying the translational motion in one dimension of a collection of molecules. You will see how the procedure is fascillated by the partition function.

Partition function: the molecules of a gas that move along one dimension can have, any of the energies given by

1676_one dimensional motion.png 

For gas samples we can assume a cubic container and express a as V1/3, where V is the volume of the sample.

The partition function for one-dimensional translational motion can be developed by recognizing that
    
The translational energy of the lowest-energy state is small compared with the energies of most of the populated states and can be set equal to zero.
    
The translational-energy spacing between successive energy levels is small compared with the range of energies of the populated states.
    
The degeneracy of each energy level is unity.

On this basis, the partition function summation over the translational energies can be replaced by integration, and the partition function is expressed as

83_one dimensional motion1.png 

The integral is one of the definite integrals dealt by using the general result shown there, we obtain

1990_one dimensional motion2.png 

Example: calculate the partition function for the translational motion of N2 molecules free to move along one dimension of a 1-L cubic container. The temperature is 25°C.

Solution: the translational-energy factor h2/(8ma2) can be calculated conveniently from the expression of this equation. The mass of M of 1 mol of N2 molecules is 0.02801 kg, and V = 1 L = 10-3 m3. Thus
2163_one dimensional motion3.png 

962_one dimensional motion4.png 

= 1.180 × 10-40 J

The value of kT, to which the energy spacing factor is compared, is

kT = (1.3807 × 10-23 J K-1) (298.15 K) = 4.116 × 10-21 J

The partition function is calculated as

1331_one dimensional motion5.png 

this large partition function value indicates that very many states are available to the molecules. This result, in the calculations, from the smallness of h2/(8ma2) compared to kT.

Average energy: the one dimensional translational energy of 1 mol of gas molecules can now be deduced. The general thermal-energy expression is

864_one dimensional motion6.png 

The partition function for one-dimensional translational motion gives
1661_one dimensional motion7.png 

substitution of the equation expressions in the equation for U - U0 gives

U - U0 = ½ RT

We have come by this long route to the result that we obtained from the simple classical kinetic-molecular theory. The translational energy per degree of freedom is ½ RT. 

   Related Questions in Chemistry

  • Q : Simulate the column in HYSYS The

    The objective of this work is to separate a binary mixture and to cool down the bottom product for storage. (Check table below to see which mixture you are asked to study). 100 kmol of feed containing 10 mol percent of the lighter component enters a continuous distillation column at the m

    		•
  • Q : Question based on vapour pressure and

    Benzene and toluene form nearly ideal solutions. At 20°C, the vapour pressure of benzene is 75 torr and that of toluene is 22 torr. The parial vapour pressure of benzene at 20°C for a solution containing 78g of benzene and 46g of toluene in torr is: (a) 50 (b)

    		•
  • Q : Haloalkene with the help of polarity of

    with the help of polarity of c-x bond show that aryl halides are less reactive than alkyl halides

    		•
  • Q : How can enzymes act as catalyst?

    Enzymes are complex proteinous substances, produced by living bodies, such as act as catalysis in the physiological reactions. The enzymes are, also called biochemical catalysts and the phenomenon is known as bio-chemical catalysis because numerous reactions that occur the bodies of animals and p

    		•
  • Q : Explain methods for industrial

    The important methods for the preparation of alcohol on large-scale are given below:    

    		•
  • Q : Explain Rotational Vibrational Spectra

    The infrared spectrum of gas samples shows the effect of rotational-energy changes along with the vibrational energy change.As we know from the interpretations given to thermodynamic properties of gases, gas molecules are simultaneously rotating and vibrating. It follows that an absor

    		•
  • Q : Adiabatic compression A lean natural

    A lean natural gas is available at 18oC and 65 bars and must be compressed for economical pipeline transportation. The gas is first adiabatically compressed to 200 bars and then isobarically (i.e. at constant pressure) cooled to 25°C. The gas, which is

    		•
  • Q : Explain the catalyst definition and

    Catalyst is a substance which accelerates the rate of a chemical reaction without undergoing any change in its chemical composition or mass during the reaction. The phenomenon of increasing the rate of a reaction with the help of a catalyst is known as catalysis.

    		•
  • Q : Molar solution of sulphuric acid Choose

    Choose the right answer from following. The molar solution of sulphuric acid is equal to: (a) N solution (b) 2Nsolution (c) N/2solution (d) 3Nsolution

    		•
  • Q : The Liver Is Responsible For Much Of

    The Liver Is Responsible For Much Of The Pentose Phosphate Activity Explain

    		•