Question

Which one of the following equations does not correctly represent the first law of thermodynamics for the given processes involving an ideal gas? (Assume non-expansion work is zero)

• A. Cyclic process: \(q = -w\)
• B. Adiabatic process: \(\Delta U = -w\)
• C. Isochoric process: \(\Delta U = q\)
• D. Isothermal process: \(q = -w\)

Hint:

In Chemistry competitive exams, always use the IUPAC sign convention (\(\Delta U = q + w\)) unless specified otherwise. In this convention, work done on the system is positive.

Answer 1

The Correct Option is B

The first law of thermodynamics is a principle concerning the conservation of energy. It is typically expressed as:

\(\Delta U = q + w\)

where:

• \(\Delta U\): Change in the internal energy of the system
• \(q\): Heat added to the system
• \(w\): Work done on the system

Let us analyze the given options:

1. Cyclic process: \(q = -w\)
   • In a cyclic process, the system returns to its initial state. Hence, the change in internal energy, \(\Delta U\), is zero.
   • Thus, \(0 = q + w \Rightarrow q = -w\).
   • This equation correctly represents the first law for a cyclic process.
2. Adiabatic process: \(\Delta U = -w\)
   • In an adiabatic process, no heat is exchanged with the surroundings, hence \(q = 0\).
   • Therefore, the first law becomes \(\Delta U = 0 + w \Rightarrow \Delta U = w\).
   • The given statement \(\Delta U = -w\) is incorrect for an adiabatic process.
3. Isochoric process: \(\Delta U = q\)
   • In an isochoric process, the volume remains constant, meaning no work is performed (\(w = 0\)).
   • Therefore, the first law becomes \(\Delta U = q + 0 \Rightarrow \Delta U = q\).
   • This equation correctly represents the first law for an isochoric process.
4. Isothermal process: \(q = -w\)
   • In an isothermal process, the temperature remains constant, leading to no change in internal energy (\(\Delta U = 0\)).
   • Therefore, the first law becomes \(0 = q + w \Rightarrow q = -w\).
   • This equation correctly represents the first law for an isothermal process.

Thus, the equation for the adiabatic process \(\Delta U = -w\) does not correctly represent the first law of thermodynamics for the given process, making it the correct answer.