recall and use the first law of thermodynamics ∆U = q + W expressed in terms of the increase in internal energy, the heating of the system (energy transferred to the system by heating) and the work done on the system

The First Law of Thermodynamics

First Law: The change in the internal energy of a closed system is equal to the heat added to the system plus the work done on the system.

In symbols: \$ΔU = q + W\$

• \$ΔU\$ – increase in internal energy (J)

• \$q\$ – heat transferred to the system (J)

• \$W\$ – work done on the system (J)

Key Concepts

  • Internal Energy (\$ΔU\$): The total energy stored in the microscopic motions and interactions of the particles in the system.

  • Heat (\$q\$): Energy transferred because of a temperature difference.

  • Work (\$W\$): Energy transferred when a force acts over a distance.

Sign Conventions

QuantityPositive WhenNegative When
\$q\$Heat added to the system 🔥Heat removed from the system ❄️
\$W\$Work done on the system 💪Work done by the system (expansion) 🚀

Analogy: The Thermos Bottle 🎈

Think of a thermos bottle as a closed system. When you pour hot coffee into it, heat (\$q\$) flows into the bottle. If you then squeeze the bottle, you do work (\$W\$) on the coffee, increasing its internal energy (\$ΔU\$). The bottle keeps the coffee warm because it limits heat loss to the surroundings.

Example Problem

Problem: A piston contains 1 mol of an ideal gas at 300 K. The gas is compressed quasi‑statically, doing 200 J of work on the gas, while 50 J of heat is transferred into the gas. What is the change in internal energy?

  1. Identify known values: \$W = +200\$ J, \$q = +50\$ J.

  2. Apply the first law: \$ΔU = q + W = 50\,\text{J} + 200\,\text{J} = 250\,\text{J}\$.

  3. Interpretation: The gas’s internal energy increased by 250 J.

Exam Tips

🔹 Remember the sign convention: Heat added → \$q>0\$; Work done on the system → \$W>0\$.

🔹 Check units: All terms in joules (J). If you see kJ, convert to J.

🔹 Use the equation in the correct form: \$ΔU = q + W\$ (not \$ΔU = q - W\$ unless you’re using the sign convention where work done by the system is positive).

🔹 Show your work: Write the equation, plug in the numbers, and state the final answer clearly.

Quick Recap

  1. Write the first law: \$ΔU = q + W\$.

  2. Determine the sign of \$q\$ and \$W\$ using the conventions.

  3. Plug values into the equation.

  4. Check that the units are consistent.

  5. State the change in internal energy.