How Do Degrees of Freedom Affect Molar Specific Heat Calculations?

[jaejoon89]

1a. Find the number of vibrational, rotational, and translational degrees of freedom of CH4 and CO2.
1b. Use the equipartition theorem to calculate the molar specific heat for hydrogen (in gas state).

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Attempt at solution:

1a.

# traslational degrees of freedom (d.o.f.) is 3 for CO2, 3 for CH4.
# rotational d.o.f. for CO2 (linear) is 2, for CH4 (nonlinear) 3
# vibrational d.o.f. for CO2 (linear) is 3N - 5 = 3(3) - 5 = 4, for CH4 (nonlinear) is 3N - 6 = 3(5) - 6 = 9

Is this correct? Do you then add them all up or multiply them or something? What does this all mean exactly?

1b.

Equipartition theorem => Cv = 5R/2

?

I really have no idea how to do this one... How do you apply the theorem here?

 

[nate808]

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Hi there!

1a. Your calculations for the degrees of freedom of CO2 and CH4 are correct. To find the total number of degrees of freedom, you simply add them up. In this case, CO2 has a total of 7 degrees of freedom (3 translational, 2 rotational, and 2 vibrational) and CH4 has a total of 15 degrees of freedom (3 translational, 3 rotational, and 9 vibrational).

1b. The equipartition theorem states that for a molecule with n degrees of freedom, the molar specific heat (Cv) is equal to (n/2)R, where R is the gas constant. For hydrogen, which has 3 translational degrees of freedom, the molar specific heat would be (3/2)R or (3/2)(8.314 J/mol*K) = 12.47 J/mol*K. This means that for every 1 mole of hydrogen gas, it would take 12.47 joules of energy to increase the temperature by 1 Kelvin.

I hope this helps! Let me know if you have any further questions.

 

[Blueshift5]

Do you need more information? Or am I completely off track?

I would like to provide a response to the above content by first acknowledging the effort put into solving the problems and then providing guidance and clarification on the concepts and solutions.

1a. The number of degrees of freedom (d.o.f.) for a molecule can be determined by the formula 3N - 6 for a nonlinear molecule and 3N - 5 for a linear molecule, where N is the number of atoms in the molecule. In the case of CO2, which is a linear molecule with 3 atoms, the number of vibrational d.o.f. is 3(3) - 5 = 4, as correctly calculated. For CH4, which is a nonlinear molecule with 5 atoms, the number of vibrational d.o.f. is 3(5) - 6 = 9, also correctly calculated. Additionally, the number of rotational d.o.f. for CO2 is 2, as it can rotate about its axis of symmetry, while CH4 has 3 rotational d.o.f. due to its asymmetric structure. The number of translational d.o.f. for both molecules is 3, as they can move freely in three dimensions. Therefore, the total number of d.o.f. for CO2 is 9, and for CH4 is 12.

1b. The equipartition theorem states that each degree of freedom in a molecule contributes 1/2RT to the molar specific heat (Cv), where R is the gas constant and T is the temperature. In the case of hydrogen gas, which has 3 translational d.o.f., 3 rotational d.o.f., and 1 vibrational d.o.f., the total number of d.o.f. is 7. Therefore, using the equipartition theorem, the molar specific heat for hydrogen gas would be Cv = 7/2R or 3.5R. However, this is only an approximation and may not be accurate for all temperatures. To calculate the molar specific heat more accurately, additional information such as the potential energy function of the molecule and the temperature range would be needed.

In summary, the concept of degrees of freedom is important in understanding the energy distribution and behavior of molecules. The equipartition theorem is a useful tool in estimating the molar specific heat of a molecule, but it may not always provide accurate