1.

Derive the ratio of two specific heat capacities of monoatomic, diatomic and triatomic molecules.

Answer» <html><body><p></p>Solution :Application of law of equipartition energy in specific heat of a gas : <br/> Meyer's relation `C_(P)-C_(Y)=R` connects the two specific heats for one mole of an ideal gas. <br/> Equipartition law of energy is used to calculate the value of `C_(P)-C_(V)` and the ratio between them `gamma=(C_(P))/(C_(V))` Here `gamma` is called adiabatic exponent. <br/> (i) Monotomic molecule: <br/> Average kinetic energy of a molecule <br/> `=[(3)/(<a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a>)kT]` <br/> Total energy of a mole of gas <br/> `(3)/(2)kTxxN_(A)=(3)/(2)RT` <br/> For one mole, the molar specific heat at constant volume `C_(V)=(dU)/(dT)=(d)/(dT)[(3)/(2)RT]` <br/> `C_(V)=[(3)/(2)R]` <br/> `C_(P)=C_(V)+R=(3)/(2)R+R=(5)/(2)R` <br/> The ratio of specific heats, <br/> `gamma=(C_(P))/(C_(V))=((5)/(2)R)/((3)/(2)R)=(5)/(3)=1.67` <br/> (ii) Diatomic molecule: <br/> Average kinetic energy of a diatomic molecule at low temperature = `(5)/(2)kT`. <br/> Total energy of one mole of gas <br/> `=(5)/(2)kTxxN_(A)=(5)/(2)RT` <br/> (Here, the total energy is <a href="https://interviewquestions.tuteehub.com/tag/purely-2962981" style="font-weight:bold;" target="_blank" title="Click to know more about PURELY">PURELY</a> kinetic) <br/> For one mole Specific heat at constant volume <br/> `C_(V)=(dU)/(dT)=[(5)/(2)RT]=(5)/(2)R` <br/> But `C_(P)=C_(V)+R=(5)/(2)R+R=(7)/(2)R` <br/> `<a href="https://interviewquestions.tuteehub.com/tag/therefore-706901" style="font-weight:bold;" target="_blank" title="Click to know more about THEREFORE">THEREFORE</a> gamma=(C_(P))/(C_(V))=((7)/(2)R)/((5)/(2)R)=(7)/(5)=1.40` <br/> Energy of a diatomic molecule at high temperature is equal to `(7)/(2)` RT <br/> `C_(V)=(dU)/(dT)=[(7)/(2)RT]=(7)/(2)R` <br/> `therefore C_(P)=C_(V)+R=(7)/(2)R+R` <br/> `C_(P)=(<a href="https://interviewquestions.tuteehub.com/tag/9-340408" style="font-weight:bold;" target="_blank" title="Click to know more about 9">9</a>)/(2)R` <br/> Note that the `C_(V)` and `C_(P)` are higher for diatomic molecules than the monoatomic molecules. It implies that to increase the temperature of diatomic gas molecules by `1^(@)C` it require more heat energy than monoatomic molecules. <br/> `therefore gamma=(C_(P))/(C_(V))=((9)/(2)R)/((7)/(2)R)=(9)/(7)=1.28` <br/> (iii) Triatomic molecule: <br/> (a) Linear molecule <br/> Energy of one mole <br/> `=(7)/(2)kTxxN_(A)=(7)/(2)RT` <br/> `C_(V)=(dU)/(dT)=(d)/(dT)[(7)/(2)RT],C_(V)=(7)/(2)R` <br/> `C_(P)=C_(V)+R=(7)/(2)R+R=(9R)/(2)` <br/> `therefore gamma=(C_(P))/(C_(V))=((9)/(2)R)/((7)/(2)R)=(9)/(7)=1.28` <br/> (b) Non-linear molecule <br/> Energy of a mole `=(6)/(2)kTxxN_(A)=(6)/(2)RT=3RT` <br/> `C_(V)=(dU)/(dT)=3R` <br/> `C_(P)=C_(V)+R=3R+R=4R` <br/> `therefore gamma=(C_(P))/(C_(V))=(4R)/(3R)=(4)/(3)=1.33` <br/> Note that <a href="https://interviewquestions.tuteehub.com/tag/according-366619" style="font-weight:bold;" target="_blank" title="Click to know more about ACCORDING">ACCORDING</a> to kinetic theory model of gases the specific heat capacity at constant volume and constant pressure are independent of temperature. But in reality it is not sure. The specific heat capacity varies with the temperature.</body></html>


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