# Internal Energy of an Ideal Gas ## Definition of Internal Energy - Internal energy of a gas is defined as: - **The sum of the kinetic and potential energies of all particles inside the gas** - For real gases, both kinetic energy (from molecular motion) and potential energy (from intermolecular forces) contribute to internal energy

Internal energy = Total kinetic energy + Total potential energy of all particles in a system

## Internal Energy in an Ideal Gas - One of the key assumptions of an ideal gas is: - **Electrostatic forces** between particles are **negligible** except during collisions - This assumption has a crucial consequence: - There is **no electrostatic potential energy** in an ideal gas - No intermolecular forces means no energy stored in interactions between molecules - Therefore, for an ideal gas: - **All internal energy is due to the kinetic energy of the particles** - Internal energy depends only on how fast the molecules are moving ## Temperature and Internal Energy - As a container of gas is heated, several changes occur: - Gas molecules move **faster** - Molecules have **higher kinetic energy** - The gas has **higher internal energy** - The relationship is direct and fundamental: - Higher temperature → faster molecules → greater kinetic energy → greater internal energy - Lower temperature → slower molecules → less kinetic energy → less internal energy ## Calculating Change in Internal Energy - The change in internal energy, $\Delta U$, equals the total kinetic energy, $E_K$, of all particles: $$\Delta U = E_K = \frac{1}{2}Nm\overline{c^2} = \frac{3}{2}Nk\Delta T$$ Where: | Symbol | Meaning | Unit | |--------|---------|------| | $\Delta U$ | Change in internal energy | J | | $E_K$ | Total kinetic energy | J | | $N$ | Number of molecules | (no unit) | | $m$ | Mass of one molecule | kg | | $\overline{c^2}$ | Mean square speed of molecules | m² s⁻² | | $k$ | Boltzmann constant ($1.38 \times 10^{-23}$) | J K⁻¹ | | $\Delta T$ | Change in temperature | K |

For an ideal gas, internal energy is directly proportional to absolute temperature: ΔU ∝ ΔT

### Visualizing the Relationship - The graph below shows how internal energy varies with temperature for one mole of an ideal monatomic gas: - Notice the **perfectly linear relationship** – this is characteristic of all ideal gases - The straight line confirms that internal energy is directly proportional to absolute temperature

## Temperature Doubling Effect - The equation demonstrates a linear relationship between temperature and internal energy - If the temperature **doubles**, the internal energy also **doubles** Example calculation: $$\Delta U = \frac{3}{2}Nk\Delta T$$ If temperature doubles ($2T$): $$\frac{3}{2}Nk(2T) = \frac{6}{2}NkT = 3NkT = 2U$$ - This shows that internal energy increases proportionally with temperature - Increasing temperature from 300 K to 600 K doubles the internal energy (3741 J → 7482 J) - Increasing temperature from 200 K to 400 K also doubles the internal energy (2494 J → 4988 J)

Temperature must always be in kelvin (K) when using the internal energy equation

## Key Points Summary - For an ideal gas: - Internal energy = kinetic energy only (no potential energy contribution) - No intermolecular forces except during collisions - All energy is in the random motion of particles - The equation $\Delta U = \frac{3}{2}Nk\Delta T$ tells us: - Internal energy depends on number of molecules ($N$) - Internal energy depends on temperature ($T$) - Internal energy is independent of pressure or volume - Doubling temperature doubles internal energy - Physical interpretation: - Hotter gas = more internal energy stored in molecular motion - Colder gas = less internal energy stored in molecular motion - At absolute zero (0 K), molecules would have minimum kinetic energy