Molar heat capacity and Degrees of freedom

In summary, the curve is smooth and leaning between the three plateaus because the equipartition theorem fails if the thermal energy that can be transferred in collisions is smaller than the energy gap between quantized energy levels.
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Homework Statement


The diagram shows the molar heat capacity of an ideal diatomic gas and the number of degrees of freedom at different temperatures. Explain why there are 3 discrete plateaus and why the curve is smooth and leaning between them.
?temp_hash=89f462dcd1470a94962b4b9e31d76d87.png


Homework Equations


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The Attempt at a Solution


For f=3 degrees of freedom we have translational motion, for f=5 we have also rotational motion and for f=7 there is also vibrational motion.

The equipartition theorem states that if a system is in equilibrium, there is an average energy of (1/2)kT per molecule or (1/2)RT per mole associated with each degree of freedom. The equipartition theorem fails if the thermal energy that can be transferred in collisions is smaller than the energy gap between quantized energy levels.

For example, the energy that can be transferred between colliding gas molecules is of the order of kT, the
typical thermal energy of a molecule. Hence in that case there are three distinct plateaus that represents the change in accordance with the quantized energy levels.

That explains why there are 3 discrete plateaus. What I wonder is why is the curve smooth and leaning between these plateaus? Could someone please explain that?
 

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A fast response to this would be appreciated :rolleyes:
 
  • #3
Effect said:
The equipartition theorem fails if the thermal energy that can be transferred in collisions is smaller than the energy gap between quantized energy levels.

For example, the energy that can be transferred between colliding gas molecules is of the order of kT, the
typical thermal energy of a molecule. Hence in that case there are three distinct plateaus that represents the change in accordance with the quantized energy levels.
Doesn't seem to me you've really explained why the number of degrees of freedom increases at certain temperature thresholds. E.g., why does rotation have a higher threshold than translation? (Btw, I don't know the answer to that.)
why is the curve smooth and leaning between these plateaus? Could someone please explain that?
Do all molecules have the same energy at once?
 

FAQ: Molar heat capacity and Degrees of freedom

What is molar heat capacity?

Molar heat capacity is a measure of the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius. It is often denoted by the symbol Cm and has units of J/mol.K (joules per mole per Kelvin).

How is molar heat capacity related to specific heat capacity?

Molar heat capacity is the specific heat capacity of a substance multiplied by its molar mass. Specific heat capacity is the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius, while molar heat capacity takes into account the larger amount of substance present in one mole.

What is the difference between molar heat capacity at constant pressure and constant volume?

Molar heat capacity at constant pressure (Cp) is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius while keeping the pressure constant. Molar heat capacity at constant volume (Cv) is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant. In general, Cp is greater than Cv because at constant pressure, the substance can expand and do work, which requires additional heat energy.

How is the number of degrees of freedom related to molar heat capacity?

The number of degrees of freedom is a measure of the number of ways a molecule can store energy. It is related to molar heat capacity through the equipartition theorem, which states that each degree of freedom contributes 1/2 kT (where k is the Boltzmann constant and T is the temperature) to the total internal energy of a substance. The more degrees of freedom a molecule has, the higher its molar heat capacity will be.

How does molar heat capacity vary with temperature?

In general, molar heat capacity increases with temperature. This is because as the temperature increases, more degrees of freedom become accessible, leading to an increase in the average kinetic energy of the molecules. However, for some substances, molar heat capacity may decrease at very high temperatures due to the onset of chemical reactions or phase changes.

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