First Law of Thermodynamics

Overview

First Law of Thermodynamics applies conservation of energy to thermal systems.

It relates internal energy, heat supplied, and work done.

Definition

Using the required convention:

$$\Delta U=Q+W$$

where $\Delta U$ is change in internal energy of the system, $Q$ is heat supplied to the system, and $W$ is work done on the system.

If using work done by the gas:

$$W_{\text{by}}=-W$$

Hence:

$$\Delta U=Q-W_{\text{by}}$$

Why It Matters

The first law explains why compressing a gas heats it, why expanding a gas can cool it, why supplied heat may not raise temperature directly, how engines convert heat into work, and why cyclic processes have zero net change in internal energy.

Key Representations

Internal energy is microscopic stored energy: random kinetic energy and intermolecular potential energy. For an ideal gas, internal energy depends only on temperature.

Heat is energy transferred because of temperature difference:

• $Q>0$: heat supplied to system;
• $Q<0$: heat leaves system.

Work $W$ is work done on the system:

• $W>0$: surroundings compress system;
• $W<0$: system expands and does work on surroundings.

If $W=0$, then $\Delta U=Q$. This applies to heating a gas in a rigid container.

If $Q=0$ and $W>0$, then $\Delta U=W$ and internal energy rises.

For fixed amount of monatomic ideal gas:

$$U=\frac{3}{2}nRT$$

so:

$$\Delta U\propto \Delta T$$

Work done by the gas from a $p$-$V$ graph is the signed area under the process curve:

$$W_{\text{by}}=\int pdV$$

Therefore:

$$W=-\int pdV$$

Expansion gives $W_{\text{by}}>0$ and $W<0$. Compression gives $W_{\text{by}}<0$ and $W>0$.

Common process results:

• isochoric: $W=0$, so $\Delta U=Q$;
• isothermal ideal gas: $\Delta U=0$, so $Q=-W=W_{\text{by}}$;
• adiabatic: $Q=0$, so $\Delta U=W$;
• cyclic: $\Delta U=0$, so $Q_{\text{net}}=-W_{\text{net}}$.

Common Exam Traps

Always check whether a question uses work done on the gas or work done by the gas.

Do not say heat is stored in a gas. Internal energy is stored; heat is transferred.

Isothermal does not mean no heat transfer. For ideal gas isothermal expansion, heat enters and equals work done by gas.

Adiabatic does not mean constant temperature. It means $Q=0$.

A cyclic process gives $\Delta U=0$, but net work can be non-zero.

For a fixed amount of ideal gas, internal energy depends on temperature, not pressure alone.

Links

• Thermal Physics B
• First Law and Thermodynamic Processes
• Temperature and Heat
• Ideal Gases
• Molecular Motion
• Thermodynamic Processes
• p-V Diagrams and Cycles
• Work, Energy and Power