All Questions
Tagged with degrees-of-freedom thermodynamics
92 questions
29
votes
2
answers
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In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted?
I was reading about the equipartition theorem and I got the following quotations from my books:
A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis down ...
user36790
12
votes
5
answers
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What are the six degrees of freedom of the atoms in a solid?
A monoatomic ideal gas has heat capacity $C_v=1.5$ which comes from the three translational degrees of freedom. For solids at high temperature, $C_v=3$, implying six degrees of freedom.
What are ...
- 389
12
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5
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Degrees of freedom and temperature
I quote the following lines directly from the Wikipedia page titled "Heat capacity":
"...rotational kinetic energy of gas molecules stores heat energy in a way that increases heat capacity, since ...
user106570
11
votes
1
answer
1k
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7/2 versus 9/2 for diatomic heat capacity
Question
I calculated the classical heat capacity of a diatomic gas as $C_V = (9/2)Nk_B$, however the accepted value is $C_V = (7/2)Nk_B$.
I assumed the classical Hamiltonian of two identical atoms ...
- 2,107
10
votes
5
answers
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Why expansion of real gases lead to cooling?
Paul Hewitt writes in his book
Expansion of real gases lead to cooling as average translational kinetic energy per molecule decreases.
The reason given is:
During Expansion molecules collide with ...
- 1,395
10
votes
2
answers
968
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Why does each mode of the electromagnetic field have two degrees of freedom?
In the derivation of the Rayleigh-Jeans law, using the Equipartition theorem the number of modes per unit frequency per unit volume is multiplied by $kT$, which implies each electromagnetic resonant ...
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7
votes
2
answers
1k
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Classical theory fails to explain quantization of motions?
I understand everything written here.
But the last point, I cannot get, at all.
How does it point towards Quantization of the two motions, since the energy change is not sudden, but gradual?
And if ...
- 205
7
votes
1
answer
259
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Specific heat capacity vs KE gain of particles
To increase the temperature of 1kg of water by 1C you need 4200J of energy. However, the KE gain is only $\frac{3}{2} k_B \Delta T \cdot 6.02\cdot 10^{23} \cdot \frac{1000}{18} = 692.3$J. Where does ...
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votes
2
answers
13k
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Why energy at room temperature $= kT$ and not $(3/2)kT$ [duplicate]
I always see that a room temperature of $T=300\,\text{K}$ corresponds to an energy of $k_BT \approx \frac{1}{40}\,\text{eV}$. But shouldn't it be $\frac{3}{2}k_BT$ since the molecules in the air have ...
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6
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2
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Gibbs phase rule and degrees of freedom at the triple point / triple line
The Gibbs phase rule tells me that at a substance's triple point, where there are 3 phases in equilibrium, there should be 0 degrees of freedom. Based on my understanding, that means there should be 0 ...
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5
votes
1
answer
3k
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How many degrees of freedom does a spring have?
I'm currently learning about thermodynamics and heat capacities. We were told that the theoretical molar heat capacities of all solids should be $3R$. I was told this is because there are 6 different ...
- 1,288
5
votes
2
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556
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What is the theoretical justification for the Law of Equipartition of Energy?
What is the theoretical justification for the Law of Equipartition of Energy?
Why are equal energies distributed in each degree of freedom even though sometimes they are completely different (like ...
user280583
5
votes
1
answer
472
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Is the energy per degree of freedom $\frac{1}{2}kT$ in relativistic systems?
The equipartition theorem says that the mean energy per degree of freedom is $\frac{1}{2} kT$. Is this result relativistically correct?
- 169
5
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2
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344
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All possible choices for two independent thermodynamic variables for a one-component, one-phase system
I am confused about the independent variables in thermodynamics. I know that for a one-component, one-phase system, there are only two independent intensive variables that can be chosen and which ...
- 474
5
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2
answers
15k
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Degrees of Freedom of a Linear Triatomic Molecule
I was introduced to a formula for finding the DOF of a molecule which was
$3N-k$ and I was told, it was just for translation and rotational degree of freedom. Here $N$ is the no. of atoms in that ...
- 409
4
votes
3
answers
591
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Why potential energy is not considered in the internal energy of diatomic molecules?
In thermodynamics, I am taught that there are 5 degrees of freedom in diatomic molecules since there are 3 for translational and 2 for rotational. I interpret degrees of freedom as "ways you can ...
- 303
4
votes
1
answer
3k
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The "potential energy" degree of freedom?
I'm reading Schroeder's "An Introduction to Thermal Physics" and he mentions the vibrational degrees of freedom of a diatomic molecule:
A diatomic molecule can also vibrate, as if the two atoms ...
- 181
4
votes
3
answers
5k
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What is the proof of $C_{V} = \frac{fR}{2}$?
I came across this formula in thermodynamics. Please give me a rigorous proof to this formula. My teacher did not even give any proof neither do any of my books. The formula is :
$C_{V}=\frac{fR}{2}$ ...
- 869
4
votes
1
answer
1k
views
Do photons have six degrees of freedom?
Calculations involving pressure and volume relationships of photon gas during the cosmologic expansion of the universe posit an adiabatic cooling process with a heat capacity ration of 4/3.
This ratio ...
- 545
4
votes
0
answers
103
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What is the deepest cause of the such high specific heat capacity of water?
Yes, I know about the hydrogen bridges. But I think, it isn't the deepest cause. Anyway, they are only second-order bindings, although quite strong.
I think, somehow should have the water a ...
- 8,338
3
votes
4
answers
2k
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Why is there a $1/2$ in the definition of energy per degree of freedom $E=(1/2)kT$?
I was looking for an authoritative definition of Boltzmann's Constant. That led me to this discussion on NIST's site: Kelvin: Thermodynamic Temperature
Thus, internal energy and temperature are ...
- 2,052
3
votes
1
answer
42k
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Degrees of freedom in a diatomic molecule [duplicate]
We know that a monatomic compound can only have 3 degrees of freedom as we can consider it to be a point mass. However now that we consider a diatomic molecule, there are 3 degrees of freedom in ...
- 427
3
votes
2
answers
986
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Why doesn't a monoatomic particle have 6 degrees of freedom? [duplicate]
A monoatomic particle can move in three directions: $x$, $y$, and $z$. So the number of degrees of freedom (DOF) for translation is 3. The particle can also rotate around three axes. So the number of ...
3
votes
1
answer
2k
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Why is the relativistic adiabatic index 4/3?
I was told that in the relativistic limit the adiabatic index approaches 4/3 for a monoatomic gas instead of 5/3 in the non-relativistic case. I was told this occurs due to a reduction in degree of ...
- 669
3
votes
1
answer
776
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The equipartion theorem and degree of freedom in case of vibration
I have been taught in chemistry that,
the energy of a vibrational freedom is $RT$ (ie, twice that of rotational/translational)
The degree of freedom which I found in chemistry, for the vibrational ...
- 63
3
votes
1
answer
939
views
How does the graph of molar heat capacity point towards quantization of motion?
In the graph below, if we consider that the jumps are stepwise for a molecule, even then, how does it point toward quantization of vibrational and rotational motion?
The thing it implies is that (...
- 205
3
votes
2
answers
1k
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Degrees of freedom in the early Universe
I am reading Dodelson's textbook on cosmology. On page 66 we find equation 3.26:
$$\rho = \frac{\pi^2}{30}T^4\biggl[\sum_{i=\text{bosons}}g_i+\frac{7}{8}\sum_{i=\text{fermions}}g_i\biggr]\equiv g_\...
- 2,134
3
votes
1
answer
1k
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Violations of Dulong-Petit rule as an upper limit to heat capacity
Does any known substance have a heat capacity at constant volume ($C_V$) per mole of atoms greater than $3k_B$ ~ 24.94 J/(mol K)?
In order to count, the substance must actually be made of atoms, that ...
- 7,787
3
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0
answers
388
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Entropy - can we express number of microstates as a function parameterized by degrees of freedom?
In some of the answers and comments from this question people contended (not in so many words) that because entropy is parameterized by number of microstates $\Omega$, and the definition of $\Omega$ ...
- 14.1k
2
votes
4
answers
48k
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For a diatomic molecule, what is the specific heat per mole at constant pressure/volume?
At high temperatures, the specific heat at constant volume $\text{C}_{v}$ has three degrees of freedom from rotation, two from translation, and two from vibration.
That means $\text{C}_{v}=\frac{7}{...
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2
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3
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937
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Why do we not consider rotation as a Degree of Freedom in Monoatomic Gases?
I completely understand that the average energy of each degree of freedom in a thermodynamic system is (1/2)kT and that we do not consider the spin about an axis of symmetry in a polyatomic molecule ...
- 67
2
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3
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1k
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Why are constant volume and constant pressure heat capacities basically the same for solids? Are degrees of freedom involved?
I knowv that $C_V=\frac{\frac{f}{2} Nk_B}{m}$ and $C_P=\frac{(\frac{f}{2} +1)Nk_B}{m}$. Since for solids their values are very close to each other, I would assume $\frac{f}{2} +1$ is very close to $\...
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2
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2
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2k
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Why does $H_2$ have $C_V$=$7/2 R$ at high temperatures, while the total number of degrees of freedom is 6?
The two hydrogen atoms have 6 degrees of freedom in total. Of them, $3$ contribute to translation, $2 $contribute to rotation and $1$ contribute to the vibration.
I know that the vibrations motion is ...
- 3,270
2
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2
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452
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Why don't we use the "degree of freedom" as a factor in the ideal gas equation?
For an adiabatic process, the ideal gas follows the equation
$$ PV^{\gamma}= constant$$
The equation above implies that the pressure of an ideal gas (under adiabatic process) depends on the "...
- 8,496
2
votes
2
answers
737
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Independent variables in thermodynamics
When we are dealing with a gaseous thermodynamic system, in books it's written that state of the system can be described by only two independent variables from the three $(p,V,T ) $. But it's not ...
- 71
2
votes
1
answer
1k
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How many degrees of freedom would water have at $\rm 500K$?
At this temperature, and lower, the rotational degrees of freedom would already be in action, then at this higher temperature id think vibration degrees of freedom are no longer frozen out. So this ...
- 433
2
votes
1
answer
3k
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Why is molar specific heat at constant volume of a monatomic ideal gas a constant?
I thought specific heat varies depending on the substance. Why is it always $(3/2) R$?
- 143
2
votes
2
answers
2k
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How many degrees of freedom has a particle in a box?
How many degrees of freedom has a particle in a rectangular box? Thing that confuses me is that box bounds the movement of the particle but to me it still seems like particle has 3 degrees of freedom.
...
- 943
2
votes
2
answers
74
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Equal average energies in translational and rotational degrees of freedom
In, An Introduction to Thermal Physics, Schroeder states
It’s not obvious why a rotational degree of freedom should have exactly the same
average energy as a translational degree of freedom. However, ...
2
votes
1
answer
212
views
Hadron contribution to effective degrees of freedom in early Universe
In transition from quark-gluon plasma to hadron gas in the early Universe, the value of the effective degrees of freedom $g_{\star}$ decreases abruptly. This seems to me like a sort of decoupling: ...
- 1,154
1
vote
4
answers
648
views
How is the relationship of the value $kT$ and a degree of freedom derived?
Sources that discuss the derivation of the Maxwell-Boltzmann Statistics end up with two unknown constants ($\alpha$ and $\beta$) through the Lagrange Multipliers, of which $\alpha$ is derived by ...
- 471
1
vote
1
answer
16k
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Why does water have 9 degrees of freedom and that too all vibrational?
How does water has 9 degrees of freedom? If it can vibrate about all three atoms then why can't a diatomic molecule also have 2 instead of 1 possible vibrations?
I haven't studied quantum mechanics ...
- 644
1
vote
2
answers
236
views
Validity of equipartition theorem and choice of coordinate axis
While reading through the basic derivation of how kinetic energy is related to temperature, I stumbled upon equipartition theorem where $\frac{1}{2}mv^2 = \frac{1}{2}kT$ thus $\frac{3}{2}kT$ in 3-...
- 434
1
vote
2
answers
684
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Degrees of freedom of a molecule
We all know that if we consider a mono-atomic molecule, it has $3$ translational degrees of freedom only, along the $3$ principal coordinates of the Cartesian coordinate system.
In case of a ...
- 1,546
1
vote
1
answer
429
views
Why is the heat capacity of water $9R$ and not $6R$?
From the equipartition theorem, the relationship between energy and temperature in a substance is $U=\frac{NRT}{2}$ for $N$ quadratic degrees of freedom associated with a particle of that substance. ...
- 21
1
vote
3
answers
214
views
Factor $f$ of internal energy of a gas
For a $n$-atomic gas in any sort of geometry,
The formula for $f$ is
$$f = 3n- \text{number of constraints}.$$
The way I was taught this formula was like each $n$ particles< there is $3$ ways it ...
- 8,040
1
vote
2
answers
996
views
Rotation About Axis of Diatomic Molecule [duplicate]
While counting the degrees of freedom of a diatomic molecule, We neglect the rotation about the axis of the molecule stating the reason that it's energy is negligible. I agree with this reasoning, and ...
- 1,108
1
vote
1
answer
92
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Vibrational degree of freedom for monoatomic gases
I read that, when the temperature of a gas becomes high enough, a third type of degree of freedom becomes accessible, viz. the vibrational degree of freedom.
Also (at high temperatures) there is a ...
1
vote
2
answers
407
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What does it mean for a degree of freedom to come to thermal equilibrium?
I'm learning about diffusion speeds of particles in aqueous solution and the fundamental concept is thermal energy.
The notes I'm working from say that "every degree of freedom comes to thermal ...
- 111
1
vote
1
answer
1k
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What is the proof of law of equipartition of energy?
In thermodynamics, law of equipartition of energy states that if we have any gas sample then the total kinetic energy will be distributed among the different degrees of freedom of the gas sample. Each ...
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