PHYS 13A Lecture

What is a mode (degree of freedom)?

An independent way for a microscopic system to store energy.

How many degrees of freedom does a monatomic ideal gas molecule have?

Three: translational motion in the x, y, and z directions.

Why does a monatomic ideal gas molecule have no rotational modes in this model?

It is treated as a point particle, so it cannot rotate.

How many degrees of freedom does a rigid diatomic molecule typically have?

Five: three translational and two rotational modes.

Why does a rigid diatomic molecule have two rotational modes?

It can rotate about two axes perpendicular to the bond; rotation about the bond is not counted in this model.

How many degrees of freedom does a nonlinear molecule such as CH₄ have in the lecture model?

Six: three translational and three rotational modes.

What are the six energy modes of a solid at high temperature in the lecture model?

Three translational kinetic modes and three potential-energy modes associated with bonds.

What does the equipartition theorem state?

On average, energy is shared equally among the independent modes of a system.

How much energy does each mode of an ideal-gas molecule have on average?

½kBT per molecule per mode.

What is kB?

Boltzmann’s constant, approximately 1.38 × 10⁻²³ J/K.

What is the average translational kinetic energy of a monatomic gas molecule?

KEavg = 3/2 kBT.

What is the internal energy of a monatomic ideal gas in terms of N?

Eint = 3/2 NkBT.

What is the internal energy of a monatomic ideal gas in terms of n?

Eint = 3/2 nRT.

What does the internal energy of an ideal gas depend on?

For an ideal gas, internal energy depends only on its temperature.

What is R?

The ideal-gas constant, approximately 8.31 J/(mol·K).

What is N in ideal-gas equations?

The total number of gas particles.

How are particle number N and amount of substance n related?

N = nNA, where NA is Avogadro’s number.

What is Avogadro’s number?

NA ≈ 6.022 × 10²³ particles/mol.

What is molar specific heat at constant volume called?

CV.

What does CV measure?

The heat required per mole to raise a gas’s temperature by 1 K while volume is held constant.

What is the heat equation for a constant-volume process?

Q = nCVΔT.

Why is work zero in a constant-volume process?

The volume does not change, so W = −pΔV = 0.

What is the internal-energy change of an ideal gas?

ΔEint = nCVΔT.

What is the general relationship between CV and degrees of freedom f?

CV = (f/2)R for the model used in the lecture.

What is CV for a monatomic ideal gas?

CV = 3/2 R ≈ 12.5 J/(mol·K).

What is CV for a rigid diatomic ideal gas with five modes?

CV = 5/2 R ≈ 20.8 J/(mol·K).

What is CV for a two-dimensional monatomic gas with two translational modes?

CV = R ≈ 8.31 J/(mol·K).

What is molar specific heat at constant pressure called?

CP.

What does CP measure?

The heat required per mole to raise a gas’s temperature by 1 K while pressure is held constant.

What is the heat equation for a constant-pressure process?

Q = nCPΔT.

How are CP and CV related for an ideal gas?

CP = CV + R.

Why is CP greater than CV for an ideal gas?

At constant pressure, some added heat expands the gas and does work, so more heat is needed for the same temperature rise.

What is the First Law of Thermodynamics using the slides’ sign convention?

ΔEint = Q + W, where Q is heat added to the system and W is work done on the system.

What is the sign of Q when heat enters the system?

Positive.

What is the sign of Q when heat leaves the system?

Negative.

What is the sign of W when work is done on the gas?

Positive.

What is the sign of W when the gas expands and does work on its surroundings?

Negative, using the slides’ convention for work done on the system.

What is the work for a constant-volume gas process?

W = 0.

What is the work for a constant-pressure gas process using the slides’ convention?

W = −pΔV.

What is the work for a reversible isothermal ideal-gas process using the slides’ convention?

W = −nRT ln(Vf/Vi).

What happens to the internal energy of an ideal gas during an isothermal process?

It does not change, because its temperature is constant.

For an isothermal ideal-gas process, how are heat and work related?

Since ΔEint = 0, the First Law gives Q = −W.

What is a state variable?

A property determined by the system’s current state, independent of the path taken to reach it.

Which quantities describe an ideal gas’s state?

Pressure p, volume V, amount of gas n, and temperature T.

Is internal energy a state variable?

Yes. Its change depends only on the initial and final states.

Are heat Q and work W state variables?

No. Each depends on the path taken between the initial and final states.

What does the First Law imply about Q + W for paths between the same two states?

Q + W is the same for every path because it equals the change in internal energy.

What is the ideal-gas law?

pV = nRT = NkBT.

What is the Celsius-to-Kelvin conversion?

T(K) = T(°C) + 273.15.

What happens to state variables over a complete cycle?

Their net changes are zero because the system returns to its initial state.

Do heat and work have to be zero over a complete cycle?

No. Q and W can each be nonzero, even though their sum gives zero net change in internal energy.