Question

2 Equipartition The laws of statistical mechanics lead to a surprising, simple, and useful result — the Equipartition Theorem. In thermal equilibrium, the average energy of every degree of freedom is the same: hEi = 1 /2 kBT. A degree of freedom is a way in which the system can move or store energy. (In this expression and what follows, h· · ·i means the average of the quantity in brackets.) One consequence of this is the physicists’ form of the ideal gas law: P V = N kBT. In this version of the law, N is the number of particles and kB is Boltzmann’s constant, kB = 1.38 · 10^−23 J/K. This can be transformed into the chemists’ version of the ideal gas law as follows: P V = N kBT = N/ NA (NAkB) T = νRT. NA = 6.02 · 10^23 is Avogadro’s number, ν is the number of moles particles in the system, and R = 8.31 J/mol · K is the ideal gas constant. The ideal gas constant is useful for describing the energy in a huge collection of particles, where as Boltzmann’s constant is useful for describing the energy of individual atoms and molecules. In this problem, we will explore the implications of the equipartition theorem. Note that kBT ≈ 4 · 10^−21 J ≈ 25 meV at room temperature.

Kinetic Energy Particles are generally free to move along three independent directions (x, y, and z, if you like). Such a particle has three degrees of freedom. What is the total kinetic energy of a particle that moves freely in three dimensions? (Express the energy in terms of vx, vy, and vz.)

Equipartition The equipartition theorem states that each of the three degrees of freedom from the previous question will have an average energy of 1/2 kBT. What is the average kinetic energy of the particle, in terms of the temperature T?

RMS Speed The average velocity of a particle that is undergoing many collisions will be zero: It is just as likely to be moving in any direction. However, its average speed may not be zero. We can use the equipartition theorem to determine the average of the square of the speed, v^2 , because it is related to the average kinetic energy: (E) = 1/2 mv^2 = 1/2 m v ^2 The square root of this quantity gives the “root mean square speed” — the rms speed: vrms =sqrt(v^2) Use the equipartition theorem to derive an expression for the rms speed of a molecule of mass m that is in thermal equilibrium with a system at temperature T.

Molecular Speed What is the rms speed of a nitrogen molecule (N2) at room temperature? Can you drive that fast?

Slow Down At what temperature would the average speed of a nitrogen molecule be 100 mph?

Thermal Noise The equipartition theorem applies to every degree of freedom in a system — every way in which it can store energy. This includes circuit elements! The energy stored in a capacitor and an inductor is U = 1/2 CV^ 2 + 1/2 LI^2 Use the equipartition theorem to find the rms voltage and rms current in a circuit that contains an inductor L = 40 mH and a capacitor C = 10 µF.

Diatomic Molecules Consider a diatomic molecule like N2. We can model this molecule as two masses attached by a spring. How many degrees of freedom does such a molecule have? Describe all of the ways the molecule can move, including rotation, as well as the potential energy stored in the spring. Draw a diagram, too.

Internal Energy What is the average thermal energy of a diatomic molecule?

Specific Heat Do you expect the specific heat of a diatomic molecule to be larger or smaller than a monatomic molecule? Explain your reasoning.

Answer #1

Q.1 (1) For one mol of a
monoatomic ideal gas in thermal equilibrium derive an expression
for the total kinetic energy of all molecules as a function of
temperature. Assume three degrees of freedom for each molecule due
to translational motion. Explain how your result is related to the
equipartition
theorem.

1. The average kinetic energy of the molecules in a gas sample
depends only on the temperature, T. But given the same kinetic
energies, a lighter molecule will move faster than a heavier
molecule.
A. What is the rms speed of Cl2 molecules at 505 K?
B. What is the rms speed of He atoms at 505 K?
2. Use the van der Waals equation of state to calculate the
pressure of 3.70 mol of H2O at 473 K in...

1-
A) A gas bottle contains 7.15×1023 Nitrogen molecules
at a temperature of 342.0 K. What is the thermal energy of the gas?
(You might need to know Boltzmann's constant: kB =
1.38×10-23 J/K.)
B) How much energy is stored in ONE degree of freedom for the
whole system?
C) What is the average energy of a single molecule?
D) On average how much energy is stored by ONE degree of freedom
for ONE single molecule?

(1) (a) Let’s derive an ideal gas law. Let’s start with a cubic
box with side-length L. Now assume we have a particle traveling
perfectly horizontally towards a single wall. When it collides with
that wall, it will turn around and hit the wall on the other side.
It will continue to bounce back and forth in this way forever. What
is the period of this motion? In other words, how much time does it
take for the particle to...

A gas bottle contains 4.90×1023 Hydrogen molecules at
a temperature of 368.0 K. What is the thermal energy of the gas?
(You might need to know Boltzmann's constant: kB =
1.38×10-23 J/K.)
Tries 0/12
How much energy is stored in ONE degree of freedom for the whole
system?
Tries 0/12
What is the average energy of a single molecule?
Tries 0/12
On average how much energy is stored by ONE degree of freedom
for ONE single molecule?

A gas bottle contains 5.40×1023 Hydrogen molecules at a
temperature of 350 K. What is the thermal energy of the gas? (You
might need to know Boltzmann's constant: kB = 1.38×10-23 J/K.)
Answer: 6.52×103 J
How much energy is stored in ONE degree of freedom for the whole
system?
What is the average energy of a single molecule?
Answer: 1.21×10-20 J
On average how much energy is stored by ONE degree of freedom
for ONE single molecule?

A gas bottle contains 9.94×1023 Hydrogen molecules at a
temperature of 339.0 K. What is the thermal energy of the gas? (You
might need to know Boltzmann's constant: kB = 1.38×10-23 J/K.)
Tries 0/20
How much energy is stored in ONE degree of freedom for the whole
system? Tries 0/20
What is the average energy of a single molecule? Tries 0/20
On average how much energy is stored by ONE degree of freedom
for ONE single molecule? Tries 0/20

Suppose that 4.8 moles of an ideal diatomic gas has a
temperature of 1061 K, and that each molecule has a mass 2.32 ×
10-26 kg and can move in three dimensions, rotate in two
dimensions, and vibrate in one dimension as the bond between the
atoms stretches and compresses. It may help you to recall that the
number of gas molecules is equal to Avagadros number (6.022 × 1023)
times the number of moles of the gas.
a) How...

Which of the following are true (can be multiple)
A. Two moles of helium (He) has the same total thermal energy as
two moles of oxygen (O) when they are both at 80° C, even though an
oxygen atom is heavier than a helium atom.
B. The average kinetic energy of an atom of an ideal gas approaches
zero at a temperature of 0° C.
C. The ideal gas law can be written as PV = NkbT,where N
is the...

1. You have two identical containers, one containing gas A and
the other containing gas B. Both gases are under the same pressure
and are at 5.0 ?C. The molecular masses are mA = 3.29 × 10?27 kg
and mB = 6.12 × 10?26 kg.
(a) (1 point) Which gas has greater translational kinetic energy
per molecule?
(b) (1 point) Which gas has greater rms speed?
(c) (1 point) Assuming you can only change one of the
containers, the temperature...

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