Chapter 2, Problem 2.92E

Interpretation Introduction

Interpretation:

H₂ gas has a single vibrational frequency of 1.295 x 10¹⁴ s^-1. The vibrational contribution to the heat capacity, C_v (vib) versus temperature from 0 to 2000 K is to be plotted.

Concept introduction:

Generally, the vibrational partition function of a molecule is derived by incorporating the calculated vibrational energy values into the exponentials and found in notation of C_v (vib) and adding everything. heat capacity (thermal capacity) is the quantity of heat required to raise the temperature of the system from the lower limit to higher divided by the temperature difference of the system. When the mass of the system is taken as 1gram, the heat capacity is denoted as specific heat capacity. Similarly, when the mass of the system taken as 1 mole, the heat capacity is referred as molar heat capacity. Heat capacity is generally described as the symbol C. Mathematically, the heat capacity of the system between two temperature T₁ and T₂ can be expressed as

$\mathrm{C}\mathrm{ }(\mathrm{T}2,\mathrm{ }\mathrm{T}1)\mathrm{ }=\mathrm{ }\mathrm{q}\mathrm{ }/\mathrm{ }(\mathrm{T}2\mathrm{ }–\mathrm{ }\mathrm{T}1)$

Answer to Problem 2.92E

The vibrational partition function is given by C_{v >}(vib) = $\frac{1}{1- e^{-\frac{h\vartheta }{kT}}}$. H₂ gas has a single vibrational frequency of 1.295 x 10¹⁴ s^-1. Therefore, C_v>(vib) at temperatures can be derived and plotted as follows.

Temperature (K)	Cv (vib)
100	≈1
500	≈1
1000	1.00202
1500	1.016
2000	1.047

Explanation of Solution

Given,

The vibrational partition function is given by C_v (vib) = $\frac{1}{1- e^{-\frac{h\vartheta }{kT}}}$

Single vibrational frequency of hydrogen = 1.295 x 10¹⁴ s^-1

We know that,

$\vartheta =1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}$

$h=6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}$

$k=1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}$

$T=500\mathrm{ }\mathrm{K}$

• C_v (vib) at 100 K:

$\frac{h\vartheta }{kT}=\mathrm{ }\frac{6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}\mathrm{ }\mathrm{x}\mathrm{ }1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}}{1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}\mathrm{ }\mathrm{ }\mathrm{x}\mathrm{ }100\mathrm{ }\mathrm{K}}\mathrm{ }$

$=62.1$

$e^{-\frac{h\vartheta }{kT} \approx 0 }$

C_v (vib) ≈1

• C_v (vib) at 500 K:

$\frac{h\vartheta }{kT}=\mathrm{ }\frac{6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}\mathrm{ }\mathrm{x}\mathrm{ }1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}}{1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}\mathrm{ }\mathrm{ }\mathrm{x}\mathrm{ }500\mathrm{ }\mathrm{K}}\mathrm{ }$

$=12.42$

$e^{-\frac{h\vartheta }{kT} \approx 0 }$

C_v (vib) ≈1

• C_v (vib) at 1000 K:

$\frac{h\vartheta }{kT}=\mathrm{ }\frac{6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}\mathrm{ }\mathrm{x}\mathrm{ }1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}}{1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}\mathrm{ }\mathrm{ }\mathrm{x}\mathrm{ }1000\mathrm{ }\mathrm{K}}\mathrm{ }$

$=6.2$

$e^{-\frac{h\vartheta }{kT} = 0.00202 }$

C_v (vib) = 1.002

• C_v (vib) at 1500 K:

$\frac{h\vartheta }{kT}=\mathrm{ }\frac{6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}\mathrm{ }\mathrm{x}\mathrm{ }1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}}{1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}\mathrm{ }\mathrm{ }\mathrm{x}\mathrm{ }1500\mathrm{ }\mathrm{K}}\mathrm{ }$

$=4.14$

$e^{-\frac{h\vartheta }{kT} = 0.0159}$

C_v (vib) = 1.016

• C_v (vib) at 2000 K:

$\frac{h\vartheta }{kT}=\mathrm{ }\frac{6.623\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-34}\mathrm{ }\mathrm{J}\mathrm{s}\mathrm{ }\mathrm{x}\mathrm{ }1.295\mathrm{ }\mathrm{x}\mathrm{ }{10}^{14}{\mathrm{s}}^{-1}}{1.38\mathrm{ }\mathrm{x}\mathrm{ }{10}^{-23}\mathrm{J}/\mathrm{K}\mathrm{ }\mathrm{ }\mathrm{x}\mathrm{ }2000\mathrm{ }\mathrm{K}}\mathrm{ }$

$=6.2$

$e^{-\frac{h\vartheta }{kT} = 0.0450 }$

C_{v >}(vib) = 1.047

Temperature (K)	Cv (vib)
100	≈1
500	≈1
1000	1.00202
1500	1.016
2000	1.047

Conclusion

Thus, the vibrational contribution to the heat capacity, C_v (vib) versus temperature from 0 to 2000 K is plotted.