Why is Cp always greater than Cv for an ideal gas?
AAt constant pressure, gas molecules move faster, so more energy is needed per degree of temperature rise
BAt constant pressure, some of the heat input does work expanding the gas against external pressure rather than raising temperature, requiring extra heat for the same ΔT
CAt constant volume, the gas loses energy through the container walls more rapidly than at constant pressure
DAt constant pressure, additional molecular degrees of freedom become accessible that are frozen at constant volume
At constant volume, all heat input goes directly into raising internal energy (and thus temperature). At constant pressure, the gas expands as it warms — this expansion does work on the surroundings (W = PΔV = nRΔT for an ideal gas), which consumes energy without raising temperature. You must supply that extra energy on top of raising the internal energy, so Cp = Cv + R. The difference is always R per mole, regardless of molecular structure — it comes from the expansion work, not from the gas's internal degrees of freedom.
Question 2 Multiple Choice
Equal moles of helium (monatomic) and nitrogen (diatomic, room temperature) are each heated by 10 K at constant volume. Which gas requires more heat?
AHelium, because lighter molecules heat up faster and need less energy, so nitrogen needs more by comparison — wait, no — helium actually needs more because monatomic gases are efficient heat absorbers
BNitrogen, because it has more active degrees of freedom (f = 5) and a higher Cv = (5/2)R compared to helium's Cv = (3/2)R
CThey require the same heat, because all ideal gases behave identically regardless of molecular structure
DHelium, because its lower molecular mass means each molecule absorbs more energy per collision
Cv = (f/2)R, where f is the number of active degrees of freedom. For monatomic helium, f = 3 (translation only), giving Cv = (3/2)R ≈ 12.5 J/(mol·K). For diatomic nitrogen at room temperature, f = 5 (3 translation + 2 rotation), giving Cv = (5/2)R ≈ 20.8 J/(mol·K). Nitrogen has more ways to store energy, so it requires more heat per degree of temperature rise. The difference in Cv directly reflects the molecular identity of the gas through the equipartition theorem.
Question 3 True / False
The heat capacities Cv and Cp of a gas depend on whether the thermodynamic process is reversible or irreversible — a reversible process has different heat capacities than an irreversible one.
TTrue
FFalse
Answer: False
Cv and Cp are state functions — properties of the gas itself, determined by its molecular structure through the equipartition theorem. They do not depend on the process. What the process determines is which heat capacity is relevant: a constant-volume process uses Cv, a constant-pressure process uses Cp. The distinction is between 'property of the gas' (Cv, Cp) and 'property of the path' (reversibility). Conflating these leads to the error of thinking Cv 'applies' only to reversible constant-volume processes.
Question 4 True / False
At very high temperatures, the molar heat capacity Cv of a diatomic gas exceeds its room-temperature value, because vibrational degrees of freedom become thermally active.
TTrue
FFalse
Answer: True
At room temperature, vibrational modes of diatomic molecules are quantum-mechanically 'frozen out' — the vibrational energy level spacing is large compared to k_BT, so few molecules are thermally excited. This gives f = 5 and Cv = (5/2)R at room temperature. At high temperatures (typically above ~1000 K for diatomic gases), vibrational modes activate, adding 2 more degrees of freedom per vibrational mode (one kinetic, one potential), pushing Cv toward (7/2)R. This temperature dependence of Cv was historically one of the first failures of classical statistical mechanics that pointed toward quantum theory.
Question 5 Short Answer
Explain why Cp = Cv + R for any ideal gas, regardless of its molecular structure.
Think about your answer, then reveal below.
Model answer: At constant pressure, heating a gas by dT requires not only raising its internal energy (which costs Cv dT per mole) but also supplying the work done by the expanding gas. By the ideal gas law, PΔV = nRΔT per mole, so the extra energy per mole per kelvin is R. Therefore Cp = Cv + R. This relationship holds for any ideal gas — monatomic, diatomic, or polyatomic — because the expansion work nRΔT depends only on the ideal gas law, not on molecular structure.
The R in Cp − Cv = R is the gas constant, which appears here because the expansion work PΔV = nRΔT is derived directly from the ideal gas law PV = nRT. This is why the difference is universal: every ideal gas expands by the same amount per mole per kelvin at constant pressure, regardless of how many internal degrees of freedom it has. The molecular structure affects Cv (through f); the ideal gas expansion adds exactly R on top of that for Cp.