Learning Materials
Features
Discover
Chapter 6: Q. 6.36 (page 246)
Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.
The average speed of the molecules of an ideal gas is given by.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
In this section we computed the single-particle translational partition function,, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula
role="math" localid="1647147005946"
where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume . By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.
Use the Maxwell distribution to calculate the average value of for the molecules of an ideal gas. Check that your answer agrees with equation 6.41.
Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.
(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.
(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.
(c) Compute the average energy again, using the formula
The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is , with . Estimate the rotational partition function for a molecule at room temperature. (Note that the arrangement of the atoms is, and the two oxygen atoms are identical.)
Estimate the partition function for the hypothetical system represented in Figure 6.3. Then estimate the probability of this system being in its ground state.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App