Class 12 > Unit # 16:First Law of Thermodynamics > Internal Energy & Temperature

Class 11 Physics • Thermodynamics

Internal Energy & Temperature

Comprehensive theory notes, microscopic kinetic vs. potential energy interpretations, and ideal gas mathematical relations.

1. What is Internal Energy?

The internal energy of a system represents the total thermodynamic energy stored within its microscopic particles.

Definition: The internal energy of a substance is defined as "the sum of the random distribution of kinetic and potential energies within a system of molecules."

• Symbol: $U$
• SI Unit: Joule ($\text{J}$)

Factors Influencing Internal Energy

The total internal energy of a physical system is determined by:

• Temperature: Governs the magnitude of random molecular motion.
• Random Motion of Molecules: Translational, rotational, and vibrational kinetic energy components.
• Phase of Matter: Intermolecular spacing affects potential energy. Gases typically possess the highest internal energy per mole, while solids possess the lowest.

Thermodynamic processes such as heating, cooling, phase transitions, and chemical reactions cause direct changes in internal energy ($\Delta U$).

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2. Relation Between Internal Energy & Temperature

The internal energy of an object is intrinsically tied to its thermodynamic temperature ($T$):

• Gases: Heating a gas increases molecular speed, directly raising random translational kinetic energy.
• Solids: Molecules are fixed in crystal lattices; heating increases the amplitude of their microscopic vibrations.
• Real Gases, Liquids & Solids: Molecules experience intermolecular forces, giving them both kinetic energy (motion) and potential energy (position/bonds).
• Ideal Gases: By definition, ideal gas particles experience no intermolecular forces. Thus, an ideal gas possesses only kinetic energy ($U_k$) and zero potential energy ($U_p = 0$).

Mathematical Relation (Ideal Gas)

Since the internal energy of an ideal gas consists purely of kinetic energy, it depends solely on absolute temperature ($T$). The change in internal energy ($\Delta U$) is directly proportional to the change in temperature ($\Delta T$):

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

$$\Delta U = \frac{3}{2} N k_B \Delta T = \frac{3}{2} n R \Delta T$$

Where:

• $\Delta U$ = Change in internal energy ($\text{J}$)
• $\Delta T$ = Change in absolute temperature ($\text{K}$)
• $n$ = Amount of gas ($\text{moles}$)
• $R$ = Universal gas constant ($8.314 \text{ J mol}^{-1}\text{K}^{-1}$)
• $N$ = Total number of gas molecules
• $k_B$ = Boltzmann constant ($1.38 \times 10^{-23} \text{ J K}^{-1}$)

Key Takeaway: As a container of gas is heated, the gas molecules move faster with higher kinetic energy, leading directly to an increase in internal energy and a proportional rise in temperature.