Question

A triatomic molecule can have a linear configuration, as does CO₂ (Figure a), or it can be nonlinear, like H₂O (Figure b). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible.

(a) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are linear?
E_int/nT =

(b) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are nonlinear?
E_int/nT =

At high temperatures, a triatomic molecule has two modes of vibration, and each contributes

1

2

R

to the molar specific heat for its kinetic energy and another

1

2

R

for its potential energy.

(c) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the linear molecules. (Use the following as necessary: R.)
E_int/nT =

(d) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the nonlinear molecules. (Use the following as necessary: R.)
E_int/nT =

(e) Explain how specific heat data can be used to determine whether a triatomic molecule is linear or nonlinear.

This answer has not been graded yet.

Are the data in table below sufficient to make this determination?

YesNo    

Answer 1

9 The energy of -The energy of one molecule can be represented as movxt movy + mo k + I wx + ₃ I WZ Its average value is 2 KBT + 2 KBT + 2 KBT + 2 KAT + 22 23T = = 58T The energy of one mole is obtained by multiplying by Avogadro's number, Eint = $ / RT and the molay heat capacity at constant volume is Eiht = 15 6) The energy of one colecule can be represented as movet many more + I w x + 2 Iwy + 2 Iw/ Its average value os BT + 2 KBT + 2 kBT + KOT + 2 KAT + 1 KBT = 3 KBT The energy of one mole is obtuned by multiplying by Avogadro's number, Eint = BRT The molar heat capacity at constant volume is Eint = [3R] c) Let the modes of vibration be denoted by 182 The energy of me molecule com be represented mm. (1f2+yutvec)+ +10 + rway + ( de vires + Kx) + (the viel + Kx) Its average value is 2 KBT + 2 KBT + 2 KAT + 2 kBT + ² krt + 2 k3T + RBT = 92 KBT T

energy of me mole is obtained by multiplying by Avogadro's number, Eint - IRT and the molow heat capacity at constant volume En't qRL The energy of one mokeule com be represented as mo (v + vytv) + ŹI wx+ / 1 wy + 2 [Wz He mu w ****) + fx evo Its average value is 3 KBT + 2 K8T + 2 kBT+ KDT + 2 137 + 2 kgT = 5KBT The energy of one mole is obtained by multiplying Avogadro's number is Fint - SRT The molar heat capacity at constant volume Eint = 5R] e) Aleasure the constant volume specific heat of the gas, as a function of temperature and look for plateaus on the graph. If the first jump goes from 3 to Sk, the molecule com be diagnosed as linear. If the first jump goes from 3 to 3R, the molecules mest be non lineds. The tabuloded data at one temperature are insufficient bes the determin nation. At room temperature, some of the heavier molecules appear to be vibrating (NO) DT