Atkins’ Physical Chemistry Eighth Edition Chapter 2 – Lecture 4 The First Law Copyright © 2006 by Peter Atkins and Julio de Paula Peter Atkins Julio de Paula

State Functions and Exact Differentials State function – state of the system independent of manner and time to prepare Path functions – processes that describe prep of the state e.g., Heat, work, temperature Systems do not possess heat, work, or temperature Advantage of state functions: Can combine measurements of different properties to obtain value of desired property

Fig 2.20 As V and T change, the internal energy, U changes adiabatic non-adiabatic When the system completes the path: dU is said to be: an exact differential q is said to be: an inexact differential

Fig 2.21 The partial derivative (∂U/∂V) T is the slope of U w.r.t V at constant T

Fig 2.22 The partial derivative (∂U/∂T) V is the slope of U w.r.t T at constant V

Fig 2.23 The overall change in U (i.e. dU) which arises when both V and T change The full differential of U w.r.t V and T

Partial derivatives representing physical properties: Constant volume heat capacity Constant pressure heat capacity Internal pressure

Fig 2.24 The internal pressure π T is the slope of U w.r.t. V at constant T has dimensions of pressure

Partial derivatives representing physical properties: Constant volume heat capacity Constant pressure heat capacity Internal pressure Now:

Two cases for real gases: Fig 2.25 For a perfect gas the internal energy is independent of volume at constant temperature

Fig 2.26 Schematic of Joule’s attempt to measure ΔU for an isothermal expansion of a gas at 22 atm q absorbed by gas ∝ T For expansion into vacuum, w = 0 No ΔT was observed, so q = 0 In fact, experiment was crude and did not detect the small ΔT

Changes in Internal Energy at Constant Pressure Partial derivatives useful to obtain a property that cannot be observed directly: e.g., Dividing through by dT: Where expansion coefficient and isothermal compressibility

Changes in Internal Energy at Constant Pressure Substituting: Into: Gives: For a perfect gas:

Changes in Internal Energy at Constant Pressure May also express the difference in heat capacities with observables: C p – C v = nR Since H = U + PV = U + nRT And: