Chapter 5: Q51E (page 190)

Air is mostly N₂, diatomic nitrogen, with an effective spring constant of 2.3 x 10³N/m, and an effective oscillating mass of half the atomic mass. For roughly what temperatures should vibration contribute to its heat capacity?

Short Answer

Expert verified

Vibration will contribute to heat capacity at temperatures of the order of 2200 K.

Step by step solution

Given data

Nitrogen has 14 protons in its nucleus. Atomic mass of Nitrogen is

$\begin{array}{rcl} m & = & 14 \times 1.67 \times 10^{- 27} kg \\ = & 23.38 \times 10^{- 27} kg \end{array}$

Spring constant is

$k = 2.3 \times 10^{3} N/m$

Difference in energy of quantum harmonic oscillator states and energy of a diatomic gas

Difference in energy of quantum states of a quantum harmonic oscillator is

$Δ E = ℏ \sqrt{\frac{k}{m / 2}}$ ,,,,,(I)

Here $ℏ$ is the reduced Planck's constant of value

$ℏ = 1.05 \times 10^{- 34} J \cdot s$

Energy of a diatomic gas is

$E = \frac{3}{2} k_{B} T$ .....(II)

Here $k_{B}$ is the Boltzmann constant of value

$k_{B} = 1.38 \times 10^{- 23} J / K$

Determining the temperature for which vibrational energy contributes to heat capacity

Equate equations (I) and (II) to get

$\begin{array}{rcl} \frac{3}{2} k_{B} T & = & ℏ \sqrt{\frac{k}{m / 2}} \\ T & = & \frac{2 ℏ}{3 k_{B}} \sqrt{\frac{k}{m / 2}} \end{array}$

Substitute the values to get

$\begin{array}{rcl} T & = & \frac{2 \times 1.05 \times 10^{- 34} J \cdot s}{3 \times 1.38 \times 10^{- 23} J/K} \sqrt{\frac{2 \times 2.3 \times 10^{3} N/m}{23.38 \times 10^{- 27} kg}} \\ = & 0.5 \times 10^{- 11} s/K \times \sqrt{0.2 \times 10^{30} \cdot (1 N \times \frac{1 kg \cdot {m/s}^{2}}{1 N}) \cdot (\frac{1}{1 m}) \cdot (\frac{1}{1 kg})} \\ = & 0.22 \times 10^{4} K \\ = & 2200 K \end{array}$

The required temperature is of the order 2200 K.

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Air is mostly N₂, diatomic nitrogen, with an effective spring constant of 2.3 x 10³N/m, and an effective oscillating mass of half the atomic mass. For roughly what temperatures should vibration contribute to its heat capacity?

Short Answer

Step by step solution

Given data

Difference in energy of quantum harmonic oscillator states and energy of a diatomic gas

Determining the temperature for which vibrational energy contributes to heat capacity

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Air is mostly N2, diatomic nitrogen, with an effective spring constant of 2.3 x 103N/m, and an effective oscillating mass of half the atomic mass. For roughly what temperatures should vibration contribute to its heat capacity?

Short Answer

Step by step solution

Given data

Difference in energy of quantum harmonic oscillator states and energy of a diatomic gas

Determining the temperature for which vibrational energy contributes to heat capacity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

Air is mostly N₂, diatomic nitrogen, with an effective spring constant of 2.3 x 10³N/m, and an effective oscillating mass of half the atomic mass. For roughly what temperatures should vibration contribute to its heat capacity?