Chapter 3: Q. 3.26 (page 108)

The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for $H_{2}$ shown in Figure 1.13, estimate the value of $ε$ for the vibrational motion of an $H_{2}$ molecule.

Short Answer

Expert verified

The value of $ε$ for the vibrational motion of an $H_{2}$ molecule is $0.44 e V$ .

Step by step solution

Given Information

The given molecule is $H_{2}$ .

Heat capacity:

role="math" localid="1647281996407" $C = \frac{ε^{2} N e^{\frac{ε}{k T}}}{k T^{2} {(e^{\frac{ε}{k T}} - 1)}^{2}}$

Calculation

The heat capacity is given as:

$C = \frac{ε^{2} N e^{\frac{ε}{k T}}}{k T^{2} {(e^{\frac{ε}{k T}} - 1)}^{2}}$

Let $t = \frac{k T}{ε}$ ,

Hence, the above equation becomes,

$C = \frac{ε^{2} N e^{\frac{ε}{k T}}}{k T^{2} {(e^{\frac{ε}{k T}} - 1)}^{2}} C = \frac{N k e^{\frac{1}{t}}}{t^{2} {(e^{\frac{1}{t}} - 1)}^{2}} \frac{C}{N k} = \frac{e^{\frac{1}{t}}}{t^{2} {(e^{\frac{1}{t}} - 1)}^{2}}$

The value of $t$ at which the heat capacity is half of its maximum value is:

$C = \frac{1}{2} N k$

Hence, the above equation becomes,

$\frac{C}{N k} = \frac{e^{\frac{1}{t}}}{t^{2} {(e^{\frac{1}{t}} - 1)}^{2}} \frac{1}{2} t^{2} {(e^{\frac{1}{t}} - 1)}^{2} = e^{\frac{1}{t}} t = 0.335$

Now, by resubstituting the value of $t$ , we get,

role="math" localid="1647282058861" $\frac{k T}{ε} = 0.335$

For vibrational motion of $H_{2}$ gas molecules at its heat capacity, half of its maximum value, the temperature is, $T = 1700 K$ .

Hence, $ε$ can be calculated as:

$ϵ = \frac{k T}{0.335} ε = \frac{8.62 \times 10^{- 5} \times 1700}{0.335} ε = 0.44 e V$

Final answer

Hence, the value of $ε$ can be calculated as $0.44 e V$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for $H_{2}$ shown in Figure 1.13, estimate the value of $ε$ for the vibrational motion of an $H_{2}$ molecule.

Short Answer

Step by step solution

Given Information

Calculation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Classical Mechanics

Torque and Rotational Motion

Modern Physics

Space Physics

Electrostatics

Study anywhere. Anytime. Across all devices.

The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for H2shown in Figure 1.13, estimate the value of εfor the vibrational motion of anH2 molecule.

Short Answer

Step by step solution

Given Information

Calculation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Classical Mechanics

Torque and Rotational Motion

Modern Physics

Space Physics

Electrostatics

Study anywhere. Anytime. Across all devices.