2. Internal Energy & Enthalpy

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Internal Energy and Enthalpy: The State Function Foundations

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1. Defining Internal Energy ($U$)

Internal energy $U$ represents the total microscopic kinetic and potential energy of all molecules in a thermodynamic system:

$$U = E_{\text{trans}} + E_{\text{rot}} + E_{\text{vib}} + E_{\text{elec}} + E_{\text{intermolecular}}$$

For a monoatomic ideal gas, only translational degrees of freedom contribute:

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

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2. Enthalpy ($H$) and Constant-Pressure Heat

When chemical reactions or biological processes occur at constant atmospheric pressure, systems expand or contract, doing work on the atmosphere:

$$H = U + PV$$

Differentiating under constant pressure ($dP = 0$):

$$dH = dU + P dV = (dq - P dV) + P dV = dq_P$$

Enthalpy change $\Delta H$ directly tracks the heat absorbed or released in open beakers and biological systems.

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Practice Problem

Problem 1: Relating C_P and C_V for an Ideal Gas

Derive Mayer’s relation $C_P - C_V = nR$ for an ideal gas.

💡 View Hint

Differentiate H = U + nRT with respect to temperature.

📝 Worked Solution & Video Walkthrough

Starting with $H = U + PV = U + nRT$:

$$\frac{dH}{dT} = \frac{dU}{dT} + nR$$

Since $C_P = \left(\frac{\partial H}{\partial T}\right)_P$ and $C_V = \left(\frac{\partial U}{\partial T}\right)_V$:

$$C_P = C_V + nR \implies C_P - C_V = nR$$