chapter ii
What is an ideal gas? | A model gas with molecules far apart and negligible intermolecular forces except during collisions.
Write the ideal gas law in terms of molecules. | pV = NkB T.
What does p represent in the ideal gas law? | Absolute pressure.
What does V represent in the ideal gas law? | Volume.
What does N represent in the ideal gas law? | Number of molecules.
What is the value of the Boltzmann constant? | 1.38 × 10⁻²³ J/K.
What temperature scale is used in the ideal gas law? | Absolute temperature (Kelvin).
Write the ideal gas law in terms of moles. | pV = nRT.
What does n represent in the ideal gas law? | Number of moles.
What is the value of R in SI units? | 8.31 J/mol·K.
What is the value of R in L·atm units? | 0.0821 L·atm/mol·K.
What is Avogadro’s number? | 6.02 × 10²³ mol⁻¹.
How are molecules and moles related? | N = nNA.
What is molar mass? | Mass of one mole of a substance.
How is molar mass related to molecular mass? | M = NA m.
What are standard temperature and pressure (STP)? | 0°C (273 K) and 1.01 × 10⁵ Pa (1 atm).
What volume does one mole of ideal gas occupy at STP? | 22.4 L.
When does the ideal gas law fail? | At high pressure or low temperature.
What equation describes real gas behavior? | The Van der Waals equation.
Write the Van der Waals equation. | [p + a(n/V)²](V − nb) = nRT.
What does the Van der Waals constant a represent? | Attractive forces between molecules.
What does the Van der Waals constant b represent? | Finite molecular volume.
What is a pV diagram? | A graph of pressure versus volume.
What is an isotherm? | A curve representing constant temperature.
What shape are ideal gas isotherms? | Hyperbolas.
What does pV equal along an ideal gas isotherm? | A constant value.
What is the critical temperature (Tc)? | The temperature above which a gas cannot be liquefied.
What is the critical point? | The point where liquid and gas phases become indistinguishable.
How do you convert Celsius to Kelvin? | T(K) = T(°C) + 273.
Why must Kelvin be used in gas laws? | Gas laws require absolute temperature.
What pressure must be used in gas calculations? | Absolute pressure.
How is absolute pressure calculated? | Gauge pressure + atmospheric pressure.
What combined gas law applies when n is constant? | p₁V₁/T₁ = p₂V₂/T₂.
When can the combined gas law be used? | When the amount of gas is constant.
What is the goal of kinetic theory? | To relate macroscopic gas properties to microscopic molecular motion.
What assumption is made about the number of gas molecules? | A gas contains a very large number of identical molecules.
What is assumed about molecular motion in a gas? | Molecules move randomly and isotropically.
What laws govern molecular motion in kinetic theory? | Newton’s laws of motion.
What assumption is made about molecular volume? | Molecular volume is negligible compared to container volume.
What type of collisions are assumed in kinetic theory? | Perfectly elastic collisions.
What does elastic collision mean? | Kinetic energy is conserved.
What equation relates pressure and molecular motion microscopically? | PV = (1/3) N m v̄².
What does v̄² represent? | The average of the square of molecular speeds.
What does the microscopic pressure equation show? | Pressure depends on molecular number, mass, and speed.
What is the formula for average kinetic energy of a gas molecule? | K̄ = (1/2) m v̄² = (3/2) kB T.
On what does average kinetic energy depend? | Absolute temperature only.
Does average kinetic energy depend on gas identity? | No.
What is the internal energy of a monatomic ideal gas? | Eint = (3/2) N kB T.
How can internal energy be written in terms of moles? | Eint = (3/2) nRT.
What type of energy makes up internal energy in a monatomic ideal gas? | Translational kinetic energy only.
What is RMS speed? | Root-mean-square molecular speed.
What is the RMS speed formula using molecular mass? | vrms = √(3kB T / m).
What is the RMS speed formula using molar mass? | vrms = √(3RT / M).
What does RMS speed estimate? | Typical molecular velocity.
What is partial pressure? | The pressure a gas would exert alone in the container.
What is Dalton’s law of partial pressures? | Total pressure equals the sum of partial pressures.
Write Dalton’s law mathematically. | Ptotal = P1 + P2 + … .
What is true about pressure-to-mole ratio at equilibrium? | P1/n1 = P2/n2.
What condition is required for Dalton’s law to apply? | Thermal equilibrium.
What is vapor pressure? | The partial pressure of a vapor in equilibrium with its liquid.
What is the dew point? | The temperature at which condensation begins.
What does relative humidity measure? | How close air is to saturation with water vapor.
Write the formula for relative humidity. | RH = (actual vapor pressure / saturation vapor pressure) × 100%.
What is the mean free path? | Average distance traveled between molecular collisions.
Write the mean free path formula using pressure. | λ = kB T / (4√2 π r² p).
What does r represent in mean free path equations? | Molecular radius.
What is mean free time? | Average time between collisions.
Write the mean free time formula. | τ = λ / vrms.
Why is mean free path longer in gases than liquids? | Gas molecules are much farther apart.
In which state is mean free path shortest? | Liquids.
What is molar heat capacity at constant volume (Cv)? | Heat required to raise the temperature of 1 mole of a gas by 1 K at constant volume.
Why is no work done at constant volume? | The volume does not change, so W = 0.
What happens to heat added at constant volume? | All heat increases internal energy.
What is the heat equation at constant volume? | Q = nCvΔT.
What is Cv for a monatomic ideal gas? | Cv = 3/2 R.
What is the value of R? | 8.31 J/mol·K.
What theorem explains molar heat capacity values? | The equipartition theorem.
What does the equipartition theorem state? | Energy is shared equally among all degrees of freedom.
How much energy does each degree of freedom contribute per molecule? | (1/2) kB T.
What is the general formula for Cv using degrees of freedom? | Cv = (d/2) R.
What are degrees of freedom? | Independent ways a molecule can store energy.
How many degrees of freedom does a monatomic gas have? | 3.
What type of motion contributes to monatomic degrees of freedom? | Translational motion only.
What is Cv for a monatomic gas using equipartition? | 3/2 R.
How many degrees of freedom does a diatomic gas have at room temperature? | 5.
What motions contribute to diatomic degrees of freedom? | 3 translational and 2 rotational.
What is Cv for a diatomic gas at room temperature? | 5/2 R.
How many degrees of freedom does a polyatomic gas have at room temperature? | 6.
What motions contribute to polyatomic degrees of freedom? | 3 translational and 3 rotational.
What is Cv for a polyatomic gas? | 3R.
What happens to degrees of freedom at very high temperatures? | Vibrational modes become active.
How many degrees of freedom does one vibrational mode add? | 2 (1 kinetic + 1 potential).
At approximately what temperature do vibrational modes activate? | Above about 3000 K.
What model describes solids in heat capacity theory? | Atoms connected by springs.
How many degrees of freedom does each atom in a solid have? | 6.
What contributes to degrees of freedom in solids? | 3 kinetic and 3 potential.
What law describes heat capacity of solids? | The law of Dulong and Petit.
What is the molar heat capacity of most solid elements? | Cv ≈ 3R.
What is the internal energy of a monatomic ideal gas? | (3/2) nRT.
What is the internal energy of a diatomic ideal gas? | (5/2) nRT.
What is the internal energy of a polyatomic ideal gas? | 3nRT.
How does internal energy depend on temperature? | It is directly proportional to temperature.
Which gas type has the smallest Cv? | Monatomic gases.
Which gas type has the largest Cv at room temperature? | Polyatomic gases.
What does the Maxwell–Boltzmann distribution describe? | The statistical distribution of molecular speeds in a gas.
What does f(v)dv represent in the Maxwell–Boltzmann distribution? | The probability that a molecule has a speed between v and v + dv.
Write the Maxwell–Boltzmann speed distribution function. | f(v) = 4π (m / 2πkB T)³ᐟ² v² e^(−mv² / 2kB T).
Why does the Maxwell–Boltzmann curve start at zero speed? | Because the probability at v = 0 is zero.
What gives the Maxwell–Boltzmann curve its initial parabolic rise? | The v² term in the distribution.
Why does the curve have a long tail at high speeds? | The exponential term decreases slowly for large v.
What is the most probable speed? | The speed at which the distribution curve reaches its maximum.
What is the formula for the most probable speed? | vp = √(2kB T / m).
What is the average (mean) molecular speed? | The arithmetic mean of all molecular speeds.
What is the formula for average speed? | v̄ = √(8kB T / πm).
What is the root-mean-square (RMS) speed? | The square root of the mean of the squared speeds.
What is the formula for RMS speed? | vrms = √(3kB T / m).
Which characteristic speed is the largest? | RMS speed.
Rank the characteristic speeds from lowest to highest. | vp < v̄ < vrms.
Which characteristic speed relates directly to kinetic energy? | RMS speed.
How does increasing temperature affect the distribution? | It shifts right and becomes broader.
Why does higher temperature broaden the distribution? | Molecules have a wider range of kinetic energies.
How does increasing molecular mass affect the distribution? | It shifts left and becomes narrower.
Why do heavier molecules move more slowly? | Speed is inversely proportional to the square root of mass.
Why is hydrogen scarce in Earth’s atmosphere? | Its high molecular speeds exceed Earth’s escape velocity.
What is Earth’s escape velocity? | Approximately 11 km/s.
What part of the distribution is important for atmospheric escape? | The high-speed exponential tail.
What is evaporative cooling? | Cooling caused by the loss of the highest-energy molecules.
Which molecules escape first during evaporation? | Molecules in the high-energy tail of the distribution.
Why does evaporation lower temperature? | Remaining molecules have lower average kinetic energy.
What does the kinetic energy distribution describe? | The probability distribution of molecular kinetic energies.
Write the Maxwell–Boltzmann energy distribution. | f(E) = (2 / √π (kB T)³ᐟ²) √E e^(−E / kB T).
How is the energy distribution related to the speed distribution? | It is obtained by converting speed to kinetic energy.
Where is the energy distribution commonly used? | Classical and quantum statistical mechanics.
What constant appears in all Maxwell–Boltzmann distributions? | The Boltzmann constant, kB.
What temperature scale must be used in these formulas? | Absolute temperature (Kelvin).