# First law, ideal gases and entropy

> Use ΔU=Q−W, where W is work done by the system. A different sign convention is possible only when used consistently.

Course: [Fields, circuits and thermal physics](https://studybrix.com/en/courses/fields-and-thermal-physics/index.md)
Level: university
Prerequisites: Calculus, pressure-volume work and temperature.

## Notes

### Convention

Use ΔU=Q−W, where W is work done by the system. A different sign convention is possible only when used consistently.

### Ideal gas

For a fixed amount of ideal monatomic gas, U=3nRT/2 in the classical translational model. Absolute temperature must be in kelvin.

### Entropy

For a reversible path, dS=δQrev/T. A state-function change may be calculated along a reversible comparison path even when the actual path is irreversible.

### Engines

Carnot efficiency is 1−Tc/Th for reversible operation between two reservoirs. Real engines need not reach that upper bound.

## Practice questions

### 1. A system absorbs 50 J and does 20 J of work. Using ΔU=Q−W, find ΔU.

Answer: 30 J

Explanation: Subtract work done by the system from heat entering it.

### 2. Which temperature scale belongs in Tc/Th for Carnot efficiency?

A. Celsius directly
B. Kelvin
C. Any shifted scale without conversion
D. Fahrenheit directly

Answer: Kelvin

Explanation: The ratio requires absolute thermodynamic temperatures.

### 3. A reversible engine operates between 600 K and 300 K. Find its Carnot efficiency as a fraction.

Answer: 0.5

Explanation: 1−300/600=0.5.

### 4. In a free expansion of an isolated ideal gas into vacuum, what is the external boundary work?

A. Infinite
B. Equal to pressure times final volume
C. Zero
D. Always positive

Answer: Zero

Explanation: The external resisting pressure is zero; isolation also makes Q zero.

### 5. For a reversible isothermal heat transfer of 40 J at 20 K, find entropy change.

Answer: 2 J/K

Explanation: ΔS=Qrev/T for this stated constant-temperature reversible transfer.

### 6. Why does Q=0 not imply ΔS=0 for every process?

Sample answer / marking guidance: An irreversible adiabatic process can generate entropy internally. Entropy transfer and entropy production are distinct.

Explanation: Credit internal entropy generation and the restriction to reversible adiabatic paths for constant entropy.

### 7. For a fixed amount of ideal gas in an isothermal process, what is ΔU?

A. Always positive
B. Zero
C. Equal to all work regardless of heat
D. Undefined because temperature is fixed

Answer: Zero

Explanation: Ideal-gas internal energy depends only on temperature.

### 8. State how a consistent sign convention helps audit a first-law calculation.

Sample answer / marking guidance: Declare whether work is done by or on the system, assign signs for each transfer, then check conservation and the physical direction of energy flow.

Explanation: Credit sign definitions and energy balance.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.