Thermodynamics — Set 1

Correct Answer: C. Q = ΔU + W

• **Q = ΔU + W** = The first law states that heat supplied to the system equals the increase in internal energy plus work done by the system. • **Sign convention** — placing W on the right means work done BY the system reduces the heat available to raise internal energy, consistent with the international standard. • 💡 Wrong-option analysis: ΔU = Q + W: this wrongly adds work to internal energy instead of subtracting it from heat; W = Q - ΔU: a valid rearrangement but not the primary statement of the law; Q = ΔU - W: the minus sign on W implies work input raises stored energy, reversing the correct convention.

Correct Answer: D. State function

• **State function** = Internal energy depends only on the current state of the system, not on the path taken to reach it. • **Path-independent** — because U is a state function, ΔU between two states is the same regardless of which process connects them. • 💡 Wrong-option analysis: Vector quantity: internal energy has magnitude only, no directional component; Dimensionless quantity: internal energy has units of joules, so it is not dimensionless; Path function: heat and work are path-dependent, but internal energy is not.

Correct Answer: D. It is zero

• **It is zero** = For an ideal gas, internal energy depends only on temperature; in an isothermal process temperature is constant, so ΔU = 0. • **ΔU = nCvΔT = 0** — since ΔT = 0 in an isothermal process, all absorbed heat is converted entirely into work done by the gas. • 💡 Wrong-option analysis: It decreases: a decrease would require the temperature to fall, but isothermal means constant temperature; It becomes maximum: there is no maximum state implied by constant temperature; It increases: an increase requires ΔT > 0, which contradicts the isothermal condition.

Correct Answer: C. Heat exchanged is zero

• **Heat exchanged is zero** = By definition an adiabatic process involves no heat transfer between the system and its surroundings, so Q = 0. • **Q = 0 → ΔU = −W** — because no heat flows, any work done by the gas comes entirely from a decrease in internal energy, causing the gas to cool. • 💡 Wrong-option analysis: Heat is rejected only: rejection is a specific direction; the defining feature of adiabatic is zero exchange, not rejection alone; Heat exchange is maximum: the opposite is true — exchange is zero; Heat is absorbed only: absorption implies Q > 0, which contradicts the adiabatic condition Q = 0.

Correct Answer: C. W = ∫ P dV

• **W = ∫ P dV** = In a quasi-static process, the gas exerts pressure P and work equals the integral of pressure over volume change. • **Area on P-V diagram** — graphically, W = ∫ P dV is the area enclosed between the process curve and the volume axis on a P-V diagram. • 💡 Wrong-option analysis: W = mgh: this is gravitational potential energy, not thermodynamic work; W = PΔT: temperature appears here instead of volume, making the units incorrect; W = ∫ V dP: this integral gives a different thermodynamic quantity related to enthalpy change, not work done by the gas.

Correct Answer: A. It is zero

• **It is zero** = In a cyclic process the system returns exactly to its initial state, so internal energy, being a state function, returns to the same value. • **ΔU_cycle = 0 → Q_net = W_net** — since ΔU = 0, all net heat absorbed in the cycle is converted to net work done by the system. • 💡 Wrong-option analysis: It equals the heat supplied: Q_net equals W_net, not ΔU; It is maximum: there is no reason for ΔU to reach a maximum in a complete cycle; It is negative: the system returns to its initial state so ΔU cannot be permanently negative.

Correct Answer: C. H = U + PV

• **H = U + PV** = Enthalpy is defined as the sum of internal energy and the product of pressure and volume, making it a state function. • **ΔH = Qp** — at constant pressure, the change in enthalpy equals the heat absorbed, which is why enthalpy is especially useful for constant-pressure processes. • 💡 Wrong-option analysis: H = PV - U: subtracting U gives a quantity without standard thermodynamic meaning; H = U + V/P: V/P has units of volume squared per force, not energy; H = U - PV: subtracting PV instead of adding it gives a negative flow direction, which is not the standard definition.

Correct Answer: D. Cp - Cv = R

• **Cp − Cv = R** = For an ideal gas, extra energy at constant pressure goes into doing P-dV work equal to R per mole per kelvin, giving Mayer's relation Cp − Cv = R. • **R = 8.314 J mol⁻¹ K⁻¹** — this universal gas constant is the exact difference, so Cp always exceeds Cv by R for any ideal gas regardless of its atomicity. • 💡 Wrong-option analysis: Cp/Cv = R: the ratio Cp/Cv equals γ, not R; Cp − Cv = 0: this would mean no extra work at constant pressure, which is impossible for an expanding ideal gas; Cp + Cv = R: the sum of heat capacities is always much larger than R, not equal to it.

Correct Answer: A. γ = Cp/Cv

• **γ = Cp/Cv** = The adiabatic index γ is defined as the ratio of the molar heat capacity at constant pressure to that at constant volume. • **γ > 1 always** — since Cp = Cv + R, γ is always greater than 1 and appears in the adiabatic equation PV^γ = constant. • 💡 Wrong-option analysis: γ = Cv/Cp: inverting the ratio gives a value less than 1 and is not γ; γ = Cp + Cv: the sum has units of J mol⁻¹ K⁻¹ and is not a dimensionless ratio; γ = Cp − Cv: this difference equals R, the gas constant, not the adiabatic index.

Correct Answer: D. Cv = 3R/2

• **Cv = 3R/2** = A monoatomic ideal gas has exactly three translational degrees of freedom; by the equipartition theorem each contributes R/2, giving Cv = 3R/2 per mole. • **Cp = 5R/2 and γ = 5/3** — adding R to Cv gives Cp = 5R/2, so the adiabatic index γ = Cp/Cv = 5/3 ≈ 1.67 for noble gases like helium and argon. • 💡 Wrong-option analysis: Cv = R: this would correspond to only two degrees of freedom, not three; Cv = 5R/2: this is Cv for a diatomic gas with five active degrees; Cv = 7R/2: this applies to a diatomic gas that also has vibrational modes active.