In this section we computed the single-particle translational partition function, $Z_{tr}$ , 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 $\frac{1}{h^{3}}$ to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula
role="math" localid="1647147005946" $Z_{tr} = \frac{1}{h^{3}} \int d^{3} r d^{3} p e^{- \frac{E_{tr}}{k T}}$
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 $V$ . By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.

Question

In this section we computed the single-particle translational partition function, $Z_{tr}$ , 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 $\frac{1}{h^{3}}$ to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula
role="math" localid="1647147005946" $Z_{tr} = \frac{1}{h^{3}} \int d^{3} r d^{3} p e^{- \frac{E_{tr}}{k T}}$
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 $V$ . By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.

Accepted Answer

The translational partition function is given by $Z_{tr} = \frac{V}{v_{Q}}$ .

Find Study Materials for

Create Study Materials

Select your language

An Introduction to Thermal Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1. Given information

Step 2. Calculation

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

An Introduction to Thermal Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1. Given information

Step 2. Calculation

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Fluids

Physics of Motion

Famous Physicists

Torque and Rotational Motion

Fields in Physics

Scientific Method Physics

Magnetism

Particle Model of Matter

Physical Quantities and Units

Medical Physics

Further Mechanics and Thermal Physics

Energy Physics

Force

Waves Physics

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Fundamentals of Physics

Kinematics Physics

Linear Momentum

Geometrical and Physical Optics

Rotational Dynamics

Electrostatics

Space Physics

Magnetism and Electromagnetic Induction

Electromagnetism

Conservation of Energy and Momentum

Turning Points in Physics

Measurements

Engineering Physics

Mechanics and Materials

Electricity

Nuclear Physics

Circular Motion and Gravitation

Oscillations

Translational Dynamics

Thermodynamics

Electric Charge Field and Potential