Training Thermodynamics Math The First and Second Laws — Energy Accounting
4 / 5

The First and Second Laws — Energy Accounting

24 min Thermodynamics Math

The first law of thermodynamics is the conservation of energy applied to thermal systems: the heat added to a system equals the change in internal energy plus the work done by the system (Q = ΔU + W). The second law introduces entropy and establishes that heat flows spontaneously only from hot to cold. Together, these two laws govern every thermodynamic process — from the expansion of steam in a turbine to the cooling of your morning coffee — and provide the framework for energy accounting in engineering design.

First and Second Laws

The laws of thermodynamics govern energy flow throughout the universe using simple algebra and inequalities.

First Law

$$\Delta U = Q - W$$

Energy is conserved. The internal energy of a system changes only by heat added or work done.

Second Law

In any spontaneous process, the total entropy of the universe increases: $\Delta S_{\text{total}} \geq 0$.

$$\Delta S = \frac{Q_{\text{rev}}}{T}$$

Example 1

A gas absorbs 500 J of heat and expands doing 200 J of work. Find $\Delta U$.

  1. $\Delta U = Q - W = 500 - 200 = 300$ J.
  2. The internal energy increases by 300 J.
Example 2

In an adiabatic process ($Q = 0$), a gas does 150 J of work. What happens to $\Delta U$?

  1. $\Delta U = 0 - 150 = -150$ J.
  2. The gas cools — its internal energy decreases.
Example 3

1000 J of heat flows from a hot reservoir at 500 K to a cold reservoir at 250 K. Find the entropy change.

  1. $\Delta S_{\text{hot}} = -1000/500 = -2$ J/K.
  2. $\Delta S_{\text{cold}} = +1000/250 = +4$ J/K.
  3. $\Delta S_{\text{total}} = -2 + 4 = +2$ J/K > 0. ✓ Entropy increases.

Practice Problems

1. $Q = 800$ J, $W = 300$ J. Find $\Delta U$.
2. Isochoric (constant V) process: $W = 0$, $Q = 400$ J. Find $\Delta U$.
3. Isothermal process: $\Delta U = 0$. If $W = 600$ J, find $Q$.
4. Entropy change when 500 J flows at 250 K?
5. Can entropy of the universe decrease spontaneously?
6. A gas expands adiabatically doing 250 J. What happens to temperature?
Show Answer Key

1. $500$ J

2. $400$ J

3. $Q = W = 600$ J

4. $\Delta S = 500/250 = 2$ J/K

5. No — the second law forbids it for spontaneous processes.

6. Temperature decreases ($\Delta U < 0$ → gas cools).

🔄 First Law Energy Balance (Q = ΔU + W)
ΔU = Q − W
Energy interpretation
Thermal efficiency (if Q>0)
Heat Engines and Carnot Efficiency Practice Test — Thermodynamics Math