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Thermodynamics and Statistical Mechanics Flashcards

Apply the laws of thermodynamics and statistical principles to calculate and understand the thermal behavior of macroscopic systems. (76 cards)

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1
Q

An ideal gas undergoes a cyclic process composed of three steps:

  1. An isothermal expansion
  2. An adiabatic compression
  3. An isochoric cooling

Determine the sign of heat, work, and internal energy for each step.

Note that work is being defined as the work done by the gas.

A
  • Isothermal: Q>0,W>0,ΔU=0
  • Adiabatic: Q=0,W<0,ΔU greater than 0
  • Isochoric: Q<0,W=0,ΔU<0

First Law is applied to each segment individually; signs follow standard conventions.

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2
Q

Is it possible for a Carnot engine to operate with 100% efficiency if the cold reservoir is at 0 K?

A

The cold reservoir cannot be at 0 K due to the Third Law of Thermodynamics.

The third law of thermodynamics implies that cooling a system to absolute zero would require an infinite number of steps or an infinite amount of time.

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3
Q

Compare the final temperature of an ideal gas that expands adiabatically vs isothermally from the same initial state to the same final volume.

A

The adiabatic expansion results in a lower final temperature than the isothermal expansion.

For isothermal expansion, the temperature remains constant.

Adiabatic processes involve no heat exchange, while in an expansion work is done by the gas. Therefore, internal energy (and thus temperature) must decrease via the first law of thermodynamics.

Note that we have a straightforward link between internal energy and temperature only for ideal gases. For real gases, the intermolecular forces have to be taken into account.

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4
Q

During a quasistatic isobaric process, the volume of an ideal gas triples.

What happens to the internal energy and entropy?

Assume ideal gas.

A

Internal energy and entropy both increase.

Temperature must rise (from PV=nRT), and both energy and entropy increase with increasing temperature (the entropy also increases because the volume increases).

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5
Q

For a gas obeying P=f(T,V), explain how one could find compressibility and thermal expansion coefficients.

A

Use thermodynamic partial derivatives.

The volume changes as the pressure changes at fixed temperature. The minus sign is so that we get a positive number.

The change in volume if we change pressure slightly is expected to be proportional to the volume, so that is why we are dividing by V - 𝛽, then tells us about the compressibility independent of what volume we are dealing with.

These quantities reflect response functions derived from the state equation P = f(T,V).

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6
Q

True or False:

The coefficient of volume expansion is approximately three times the coefficient of linear expansion for isotropic materials.

A

True

For isotropic materials, the volume expansion coefficient β is approximately three times the linear expansion coefficient α, i.e., β = 3α, due to the additive nature of expansion in three orthogonal directions.

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7
Q

Why is entropy considered a signature of irreversibility in thermodynamic systems?

A
  • Δ𝑆 = 0 only for a reversible process.
  • For an irreversible process, the total entropy always increases.

Real processes, which are inevitably irreversible, increase the total entropy (entropy of system plus entropy of surroundings); reversible ones keep it constant.

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8
Q

A monatomic ideal gas undergoes an adiabatic compression from volume V₁ to V₂.

Derive the final pressure.

A

From PVγ =const for adiabatic ideal gases.

  • To derive this equation, use the first law of thermodynamics as well as the ideal gas law.
  • The former leads to nCv dT = - p dV
  • Note that for an ideal gas, the change in internal energy is written in terms of Cv even when the volume is changing.
  • Use p = nRT/V to get rid of p
  • Separate variables and integrate.
  • Recognize that R/Cv is γ - 1 for an ideal gas.

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9
Q

In a sealed vertical cylinder with a frictionless piston, heating an ideal gas causes the piston to rise slowly. What process occurs?

Assume the piston is frictionless and the external pressure remains constant.

A

Isobaric expansion

To be more precise, this is quasi-static isobaric expansion.

Constant external pressure means that the external pressure (due to atmospheric pressure and the weight of the piston) remains approximately equal to the pressure inside the cylinder as the piston rises slowly.

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10
Q

Why is the internal energy of an ideal monatomic gas proportional to temperature?

A

For the ideal gas, the intermolecular forces are ignored, so the internal energy can only depend on the average kinetic energy.

As we increase the temperature, the kinetic energy increases. More precisely, the internal energy is U=(3/2)nRT.

Internal energy arises solely from translational motion in ideal monatomic gases.

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11
Q

Two containers of equal volume and temperature contain the same number of helium and argon atoms. Which has higher pressure, assuming that both can be treated as ideal gases?

A

Both exert the same pressure.

Pressure depends only on n,T,V; particle mass is irrelevant in ideal gases.

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12
Q

What are the main types of statistical ensembles, and what quantities remain fixed in each?

A
  • Microcanonical (N, V, E): Fixed number of particles, volume, energy.
  • Canonical (N, V, T): Fixed number of particles, volume, temperature.
  • Grand canonical (μ, V, T): Fixed chemical potential, volume, temperature.

Each ensemble corresponds to different physical constraints. In the thermodynamic limit, each of these ensembles leads to the same results.

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13
Q

In the canonical ensemble, how is entropy related to the partition function Z?

A

Derived from the Helmholtz free energy F = -kB T ln Z.

Also, the average energy is the derivative of ln Z with respect to inverse temperature, and the entropy is essentially the derivative of F with respect to temperature. These are partial derivatives.

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14
Q

Explain why the microcanonical ensemble is appropriate for isolated systems.

A

Because it assumes fixed E,V,N - we have an isolated system, so these quantities do not change.

We then have equal probability for all accessible states.

The entropy is maximized with the constraints of fixed E,V,N. Equal probability then maximizes the entropy.

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15
Q

What is the relationship between entropy and the number of accessible microstates in the microcanonical ensemble?

A

W is the number of accessible microstates.

This expression is fundamental to statistical mechanics, establishing a direct link between microscopic configurations and macroscopic thermodynamic properties.

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16
Q

Discuss the role of the chemical potential in the grand canonical ensemble.

A

Physically, the chemical potential tells us:

  • the change in energy when one particle is added to a system at constant entropy and volume
  • two systems that can exchange particles are in equilibrium when their chemical potentials are equal
  • A higher chemical potential means that the system would like to have more particles.

In the grand canonical ensemble, the particle number is not fixed. What is fixed instead is the chemical potential.

The chemical potential is a crucial variable in describing open systems, influencing phase transitions and reaction equilibria.

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17
Q

What is the significance of the Helmholtz free energy (F) in the canonical ensemble?

A

Knowing F allows us to determine all the other thermodynamic quantities, such as the entropy.

At fixed volume and temperature, equilibrium corresponds to minimization of F.

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18
Q

A rigid container with a single gas has a small hole. As temperature increases, how does mass loss due to effusion vary?

A

Rate increases due to higher average molecular speed.

Higher speed means that the molecules of the gas strike the hole more often.

As temperature is increased, the average molecular speed increases.

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19
Q

Why do solids expand with temperature, despite strong intermolecular forces?

A

The potential energy curve describing the interaction of two atoms within a solid is not exactly symmetric about the equilibrium position.

  • Atoms are vibrating about their equilibrium positions.
  • At smaller distances, the restoring force is greater than at distances longer than the equilibrium length.
  • This means that on average, the atoms spend more time when their separation is bigger than the equilibrium distance.
  • As the solid is heated, the atoms vibrate more, so their average separation increases since they spend more time further away from each other.

Anharmonicity in atomic potential leads to thermal expansion.

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20
Q

A rod of length L has ends maintained at different temperatures.

Derive the steady-state temperature profile.

A

T1: The temperature at the cold end
T2: The temperature at the hot end

The steady-state temperature is found by solving Laplace’s equation. Laplace’s equation admits no minima or maxima in its solutions. Moreover, given the boundary conditions, the solution has to be unique. The given solution satisfies Laplace’s equation (double derivative of T with respect to x is zero) as well as the boundary conditions.

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21
Q

Explain why placing reflective insulation reduces heat transfer in a house.

A

Reflective surfaces reduce radiative heat transfer by reflecting radiation rather than absorbing it.

Reflective insulation has low emissivity. This means that it radiates minimally to its surroundings; rather, it reflects most of the incoming radiation.

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22
Q

Derive the expression for the change in entropy for an ideal gas undergoing an isothermal expansion from volume Vi to Vf.

A

For an isothermal process, the change in internal energy is zero. Then T dS = p dV.

Using p = nRT/V, integrate both sides, and use the fact that T is constant for the isothermal process.

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23
Q

True or False:

The efficiency of a Carnot engine depends only on the temperatures of the heat reservoirs.

A

True

The efficiency (η ) of a Carnot engine is given by η = 1 - (Tc/Th), where Tc and Th are the absolute temperatures of the cold and hot reservoirs, respectively.

No engine can have efficiency greater than the efficiency of a Carnot engine (for the same heat reservoirs).

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24
Q

Identify the thermodynamic potential that is minimized in a closed system at constant temperature and volume.

A

Helmholtz free energy (F)

For a system at constant temperature and volume, the Helmholtz free energy ( F = U - TS ) is minimized at equilibrium.

This can be derived by looking at the interaction of the system and a surrounding large heat bath whose temperature effectively remains fixed. Note that a ‘closed system’ means that the system can exchange energy with its surroundings but not particles.

We then impose the condition that the total entropy cannot decrease. This leads to the minimization of F. Note that we assume that the system is not capable of performing electrical or other ‘non-pV’ work.

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25
What thermodynamic quantity will be **minimized** at constant energy and volume?
The **negative** of the entropy. ## Footnote According to the second law, the entropy S is maximized. That's mathematically the same as saying that -S is minimized.
26
Identify the **thermodynamic potential** most relevant to **closed systems** at constant temperature and pressure.
Gibbs Free Energy ## Footnote For constant temperature and pressure, the Gibbs free energy is minimized. The derivation is very similar to how we are led to the conclusion that the Helmholtz free energy is minimized at constant volume and temperature, except that since the volume is not constant, we have to take this 'pV work' into account as well.
27
Describe the role of **Gibbs free energy** in determining the spontaneity of a process at constant pressure and temperature.
G=H−TS = U + pV - TS | A process is spontaneous if ΔG<0. ## Footnote Minimizing Gibbs energy at constant T,P determines equilibrium and spontaneity. If the system is not in equilibrium, then the total entropy of the Universe increases if the Gibbs free energy of the system decreases. The system evolves to reduce its Gibbs free energy without requiring any external intervention.
28
# Fill in the blank: In a reversible process, the **total entropy** change of the system and its surroundings is \_\_\_\_\_\_.
zero ## Footnote Reversible processes are idealized processes in which the system and surroundings can be returned to their original states without any net change in entropy. For irreversible processes, the entropy would increase.
29
Derive the **Clausius inequality** and explain its significance in thermodynamics.
## Footnote This expresses the Second Law: entropy increases in real (**irreversible**) processes. This inequality allows us to say that the entropy change from state A to state B is greater than (or equal to) the integral of ΔQ/T from A to B.
30
# True or False: * A system accepts 200 J of heat from a bath with temperature 400 K. * The bath is then disconnected from the system and its temperature is made 300 K. * The system is again put into thermal contact with the bath. * The system now loses 150 J of heat to the bath. * The initial and final states of the system are the same. **The system has undergone a reversible cycle.**
True ## Footnote ∮𝑑𝑄/𝑇= 200/400−150/300=0
31
A heat engine absorbs heat QH from a hot reservoir and expels heat QC to a cold reservoir. **Derive the efficiency.**
## Footnote Efficiency is ratio of work output to 'input' heat: W=QH−QC
32
# True or False: A gas expands isothermally and then is compressed adiabatically to its initial volume. The change in internal energy is zero.
False ## Footnote During the expansion, the change in internal energy is zero since the expansion is isothermal. However, for the adiabatic compression, work is done on the gas, leading to an increase in the internal energy of the gas. The initial and final states of the gas are not the same. While they have the same volume, they differ in temperature.
33
State the **Zeroth Law of Thermodynamics**.
If two systems are each in **thermal equilibrium** with a third system, they are in thermal equilibrium with each other. ## Footnote It establishes that there must be something that we call 'temperature' and it helps us in defining scales for this.
34
Derive the expression for **work done in an isothermal expansion** of an ideal gas.
## Footnote From W=∫PdV and P=nRT/V for isothermal ideal gas.
35
Derive the expression for the work done in an **adiabatic compression** of an ideal monatomic gas in terms of the initial temperature Ti and final temperature Tf.
## Footnote * Use pVgamma = constant = piViγ. * Then p = pi (Vi/V)gamma. * Use this in W = ∫PdV. * Use pV = nRT to write the final answer in terms of temperatures. Note that γ = 5/3 for an ideal monatomic gas.
36
# True or False: In an **adiabatic process**, the change in the entropy of a system is zero.
False ## Footnote This is only true for a reversible process. For an irreversible process, the entropy changes. See the second law of thermodynamics. Physically, friction can, for example, cause irreversibility. The presence of friction will increase the entropy.
37
In the **free expansion of an ideal gas**, what are the changes in internal energy and the entropy?
* The change in internal energy is zero. * The entropy increases. ## Footnote The change in internal energy is zero because there is no heat exchange and no work is done. The entropy increases because the expansion is an irreversible process.
38
Derive the expression for the **pressure of an ideal gas from kinetic theory.**
1. Pressure from kinetic theory is P= (1/3)ρv², where ρ = Nm/V. 2. Relate the rms speed of the molecules to the temperature at (3/2)kBT 3. Then, we arrive at PV = nRT, the ideal gas law. ## Footnote Derived from analyzing the molecular collisions with the walls and averaging over the velocities. R is defined in terms of the Boltzmann constant and Avogadro's number.
39
Explain how to calculate the **heat required to convert a substance from solid to vapor**. Consider **only** the heat required for the phase transitions to take place.
## Footnote Where Lf is latent heat of fusion and Lᵥ is latent heat of vaporization. Heat added during phase changes increases potential energy, not kinetic energy. Thus, temperature stays constant during a phase transition.
40
# True or False: The **internal energy change** of an ideal gas in a cyclic process is zero.
True ## Footnote In a cyclic process, **the system returns to its initial state**, so the change in internal energy, a state function, is zero regardless of the path taken.
41
Explain why real gases deviate from ideal behavior at **high pressures**.
* Intermolecular forces become significant. * Volume occupied by gas molecules is non-negligible. ## Footnote At high pressures, the **assumptions of negligible intermolecular forces and molecular volume in the ideal gas law break down**, requiring corrections such as those provided by the van der Waals equation.
42
Describe the role of the **partition function** in statistical mechanics.
* Encodes all thermodynamic information of a system. * **Determines properties** like energy, free energy, entropy, and heat capacity. * Serves as a bridge between microscopic states and macroscopic quantities. To calculate the partition function, we sum over the microscopic energy states. ## Footnote The partition function is a central concept in statistical mechanics and is crucial for deriving thermodynamic properties from statistical considerations.
43
# True or False: The **ideal gas** law is a special case of the **van der Waals equation.**
True ## Footnote The ideal gas law assumes no interactions between particles and that the volume of particles is negligible. The corrections to the ideal gas law provided by the van der Waals equation become important at low temperatures and high pressures, since in these regimes the assumptions of no interactions and negligible volume of the particles become questionable.
44
Find the conditions for the critical point using the **van der Waals equation**.
## Footnote At the critical point, the isotherm has a point of inflection. These conditions allow us to find that Vc = 3nb, Tc = (8a) / (27Rb) and Pc = a / (27b²).
45
For a system of particles in 3D bound by potential U∝rⁿ, use the **virial theorem** to **relate average kinetic and potential energy.**
## Footnote The virial theorem says that 2⟨T⟩ = ⟨r · ∇U(r)⟩. For our case, 2⟨K⟩ = n⟨U⟩ (for the potential U∼rⁿ). This follows from the virial theorem since r · ∇U(r) = nU(r) for the given form of U.
46
A system undergoes a **first-order phase transition** at constant temperature and pressure. What happens to **entropy**?
## Footnote L is the latent heat. The entropy increases if the system goes from a more ordered phase to a less ordered one.
47
During a first-order phase transition, why does **temperature remain constant** while heat is added?
The energy goes into **changing the phase**, not increasing temperature, so latent heat is absorbed. ## Footnote Heat contributes to breaking intermolecular bonds rather than increasing kinetic energy.
48
Explain the role of the **Gibbs free energy** in determining the coexistence of phases at equilibrium.
At equilibrium, Gibbs free energy per mole is equal across phases. ## Footnote G determines spontaneity at constant P and T; phase coexistence occurs when ΔG=0. There is no net driving force for the system to convert entirely into one phase or the other.
49
What is the **Clapeyron equation** for phase equilibrium?
## Footnote This equation relates slope of coexistence curve to entropy and volume changes. * This can be derived by looking at the equality of the Gibbs free energies, where dG = 0 for both phases. * Note that dG = V dP - S dT. * Write this for both phases * Use dG = 0 for both phases, and rearrange. * At the end, we also use that the change in entropy is related to the latent heat.
50
Near a **second-order phase transition**, why do response functions (like heat capacity) diverge?
Due to the emergence of **long-range correlations** and **critical fluctuations**. ## Footnote Mean-field theories near the critical point break down. The second derivatives of the free energy can diverge.
51
# Fill in the blank: A **monoatomic ideal gas** has \_\_\_\_\_\_ degrees of freedom.
3 ## Footnote We have translational motion in three independent directions. There are no rotational or vibrational modes since the gas is monatomic.
52
# Fill in the blank: The **ratio of specific heats** ( γ = Cp/Cv) for an ideal monatomic gas is \_\_\_\_\_\_.
5/3 ## Footnote For a monatomic ideal gas, Cv = (3/2)R and Cp = Cv + R = 5/2R, leading to γ=5/3=1.67. Note that the relation Cp = Cv + R is not true if the gas is not ideal. Also, Cv = (3/2)R is only true for ideal monatomic gases.
53
What is the most probable speed of particles in a gas of **mass m at temperature T**?
## Footnote This is the peak of **Maxwell–Boltzmann** speed distribution. This is NOT the same as the root mean square speed. The rms speed is higher than this because of the tail (in the Maxwell-Boltzmann distribution) of fast-moving particles.
54
How does the **Maxwell-Boltzmann distribution** change with temperature, and what implications does this have for the behavior of gases?
* As temperature increases, the distribution broadens (a wider distribution of speeds) and shifts towards higher speeds. * Higher temperatures lead to an increased average speed and the kinetic energy of the particles. ## Footnote This affects diffusion rates and reaction kinetics, embodying the temperature dependence of chemical processes.
55
How does the **Maxwell-Boltzmann distribution** explain the rate of evaporation in a liquid at a given temperature?
Molecules in the **high-velocity tail** of the distribution can overcome the intermolecular forces at the surface of the liquid and escape the liquid. ## Footnote Even at moderate temperatures, some particles have **enough kinetic energy to vaporize** since the particles have a range of speeds.
56
Calculate the mean **free path** of an ideal gas molecule and explain its dependence on pressure and temperature.
## Footnote Inversely with P, directly with T. This result is derived using kinetic theory. As temperature increases, the molecules zoom about more quickly, meaning that they travel farther before colliding again. It's inversely proportional to pressure since increasing pressure means that the molecules are squeezed together more, thus they collide sooner.
57
Explain the principle of **equipartition of energy**.
The equipartition theorem says that each quadratic **degree of freedom** contributes (1/2)kT of energy per particle. ## Footnote For example, in a monoatomic gas, the 3 translational degrees of freedom (motion in x, y, and z) each contribute (1/2)kT, which results in a total average energy of (3/2)kT. The equipartition theorem is valid in the classical regime; it breaks down at low T due to quantum effects.
58
# True or False: For an **ideal gas** undergoing an adiabatic process, the equation PVγ = constant holds.
True ## Footnote This relation is derived from the first law of thermodynamics for adiabatic processes where dU = -p dV. For an ideal gas, the change in internal energy only depends on the change in temperature, not changes in temperature and volume.
59
# True or False: The **mean free path** in a gas is inversely proportional to the square root of the temperature.
False ## Footnote The **mean free path** is actually proportional to the temperature. At higher temperatures, the gas molecules travel a longer distance before colliding due to their (on average) higher speed.
60
# True or False: In the **microcanonical ensemble**, the temperature of the system is constant.
False ## Footnote The microcanonical ensemble is characterized by constant energy, volume, and number of particles (NVE ensemble), not constant temperature. Its primary use is in describing isolated systems. It can be considered to be the fundamental ensemble in statistical mechanics since any system becomes isolated if we define our system to be 'big' enough.
61
# Fill in the blanks: The **grand canonical ensemble** is used to describe systems that can exchange both \_\_\_\_\_\_ and \_\_\_\_\_\_ with a reservoir.
energy; particles ## Footnote This ensemble is characterized by constant chemical potential, volume, and temperature (μVT ensemble) and is particularly useful in quantum statistical mechanics when dealing with identical particles.
62
Explain the concept of **ergodicity in statistical mechanics**.
Ergodicity implies that **time averages** are equal to ensemble averages for a system's observables ## Footnote Ergodicity is a foundational assumption that connects microscopic dynamics with macroscopic thermodynamic properties, allowing predictions of equilibrium behavior. It justifies replacing **time averages with statistical averages.**
63
Under what conditions does the **Gibbs paradox arise**, and how is it resolved?
* Arises in classical statistical mechanics when treating **identical particles as distinguishable**. * In particular, we look at the mixing of two identical ideal gases. A classical calculation predicts an entropy increase, which should not be the case because the gases are identical—nothing changes physically after mixing since the gases are identical. * Resolved by dividing the phase space volume by N!, where N is the number of particles. ## Footnote The resolution acknowledges the indistinguishability of identical particles, which is a key concept in quantum mechanics and affects how quantum states are counted.
64
Compare the use of **classical and quantum statistics** in describing **high-density systems.**
* Classical statistics (Boltzmann) breaks down at high densities due to indistinguishability and wave nature of particles. * If the particles are relatively far apart (low density systems) these effects play less of a role, the wavefunctions barely overlap and we can tell which particle is which. * Quantum statistics (Bose-Einstein, Fermi-Dirac) must be used for high density systems. ## Footnote Quantum statistics account for the effects of quantum indistinguishability and the Pauli exclusion principle, crucial for systems like electron gases and photon gases. These high density systems are not very exotic - to understand light or electrons in a metals, the particles have to treated as identical quantum particles.
65
What is **enthalpy (H)** and how is it defined?
H=U+PV ## Footnote It is useful at constant pressure processes. An example of a constant pressure process would be a typical chemical reaction. * Under these conditions, the first law of thermodynamics simplifies to Q = Δ H. * This is because at constant pressure Δ H = Δ U + PΔ V, leading Δ H = Δ U + W, which by the first law of thermodynamics, becomes ΔH = Q. * So ΔH represents the heat absorbed or released in a chemical reaction. If it is *positive*, it means that the system absorbed heat -- an endothermic reaction. If it is *negative*, the system released heat to surroundings in an exothermic reaction.
66
Under **what conditions** is ΔH=Q?
At **constant pressure** in a process with only PV work. ## Footnote For example, there is no electrical work. When Q=ΔH; a situation that is common in chemistry and biology.
67
Explain the significance of the **Gruneisen parameter** in the context of thermal expansion and its relation to anharmonicity in a crystal lattice.
* A solid expands as the temperature is increases due to anharmonicity - about equilibrium, the restoring force is not symmetric in magnitude. * The Gruneisen parameter ( γ ) relates volume changes to the vibrational properties of a crystal lattice. * It indicates the degree of anharmonicity in the lattice vibrations, which affects thermal expansion. * A higher ( γ ) suggests greater anharmonicity and larger thermal expansion. ## Footnote Anharmonicity refers to deviations from Hooke’s law in the potential energy of the lattice and influences thermal expansion properties.
68
The heat equation says that ∂u/∂t = α ∇²u. where u is the temperature field. **Explain briefly what this equation means in simple terms.**
* The Laplacian of the temperature measures the difference between the value of the temperature at a point (for a given value of time) and its average value. * This is computed by looking at its value at neighboring points (at the same value of time). * If these two values are not equal, then temperature must change (heat 'diffuses') and the rate of this change is proportional to the difference.
69
# True or False: The **heat equation** is derived assuming constant thermal conductivity, density, and specific heat capacity of the material.
True ## Footnote The classical heat equation assumes constant thermal properties to simplify the analysis, resulting in a linear partial differential equation. Variations in these properties necessitate more complex, often non-linear, models.
70
A rod with thermal conductivity k and cross-sectional area A conducts heat between ends at T1 and T2. **Derive the heat flow.**
## Footnote Follows from **Fourier’s law** of conduction. This states that the rate of heat flow Q through the rod is given by Q = -kA dT/dx. This assumes that the temperature changes linearly over the length of the rod.
71
Why is the **molar heat capacity** at constant pressure Cp greater than Cv for an ideal gas?
Because **work** is done during expansion: Cp−Cv =R. ## Footnote * At constant volume, Q = ΔU (no work is done). * At constant pressure, Q = Δ U + W. (some heat goes into doing work) * For an ideal gas, ΔU depends only on temperature. * For the same temperature change, Q is greater at constant pressure, making Cp > Cv
72
A solid obeys the **Dulong–Petit law** at high temperatures. How does this affect its molar heat capacity at constant volume?
It approaches CV =3R. ## Footnote Classical equipartition gives 3 translational + 3 vibrational degrees of freedom, each with (1/2)kT per quadratic mode. Each atom can be considered to be a three-dimensional harmonic oscillator. As such, we have six quadratic degrees of freedom per atom.
73
Why does the **heat capacity of diatomic gases** deviate from the classical value at low temperatures?
Quantum effects suppress rotational and vibrational modes that are classically expected to contribute. This is due to quantization of energy. **At low temperatures, the thermal energy is not sufficient to cause excitation to some energy levels.** ## Footnote Only translational modes contribute significantly at low T, reducing the heat capacity below classical predictions.
74
A **blackbody surface** radiates at 600 K. If its temperature doubles, how does its **radiated power** change?
Increases by a factor of 16. ## Footnote P∝T⁴ ; doubling T gives 2⁴=16 times more power.
75
For photons in a cavity, what is the **chemical potential**?
zero ## Footnote The number of photons is not fixed. As an example, consider the light in your room. If you turn off the light sources, the number of photons obviously goes down. Since the number of photons is not fixed (photons can be freely created or destroyed in thermal equilibrium), the chemical potential must be zero since the chemical potential is the energy cost of adding a particle.
76
Consider a quantum mechanical harmonic oscillator. What is the **partition function**? What is the **average energy**?
## Footnote For Z, we then get an infinite geometric series that we can sum over.
GRE® Physics
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