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The First and Second Laws — Energy Accounting

24 min Thermodynamics Math

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 $\Delta U$.

$\Delta U = Q - W = 500 - 200 = 300$ J. 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 $\Delta U$?

$\Delta U = 0 - 150 = -150$ J. 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.

$\Delta S_{\text{hot}} = -1000/500 = -2$ J/K.

$\Delta S_{\text{cold}} = +1000/250 = +4$ J/K.

$\Delta S_{\text{total}} = -2 + 4 = +2$ J/K > 0. ✓ Entropy increases.

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).