1. 1 First Law -- Part 2 Physics 313 Professor Lee Carkner Lecture 13

2. 2 Ideal Gas At low pressure all gases approach an ideal state lim (PV) = nRT The internal energy of an ideal gas depends only on the temperature: (dU/dP) T = 0 (dU/dV) T = 0 The first law can be written in terms of the heat capacities: dQ = C V dT +PdV dQ = C P dT -VdP

3. Heat Capacities Heat capacities defined as: C V = (dQ/dT) V = (dU/dT) V C P = (dQ/dT) P Heat capacities are a function of T only for ideal gases: Monatomic gas c V = (3/2) R c P = (5/2) R Diatomic gas c V = (5/2) R c P = (7/2) R  = c P /c V

4. 4 Adiabatic Process For isothermal, isobaric and isochoric processes, something remains constant –What remains constant for an adiabatic process? Can calculate from first law and dQ=0

5. 5 Adiabatic Relations dQ = C V dT + PdV dQ = C P dT -VdP VdP =C P dT PdV = -C V dT (dP/P) = -  (dV/V) PV  = const. Plotted on a PV diagram adibats have a steeper slope than isotherms

6. 6 Ruchhardt’s Method How can  be found experimentally? –Need to vary P and V adiabatically Ruchhardt used a jar with a small oscillating ball suspended in a tube Motions of the ball caused adiabatic expansion and contraction The period of simple harmonic motion related to pressure and volume

7. 7 Finding  The pressure and the volume changes are related to the force and displacement Also related to PV  and Hooke’s law  = (4  2 mV)/(A 2 P  2 ) –Where  is the period Modern method uses a magnetically suspended piston (very low friction)

8. 8 Microscopic View Classical thermodynamics deals with macroscopic properties –P,V and T are measured directly –Equations of state are determined by experiment What causes P, V and T? The microscopic properties of a gas are described by the kinetic theory of gases

9. 9 Kinetic Theory of Gases The macroscopic properties of a gas are caused by the motion of atoms (or molecules) –Temperature is related to the velocities and kinetic energy of the atoms –Pressure is the momentum transferred by atoms colliding with a container –Volume is the space occupied by moving atoms Consider a monatomic ideal gas

10. 10 Assumptions Any sample has large number of particles (N) –Can treat them statistically Atoms have no internal structure –Behave like marbles in motion No forces except collision –Ignore chemistry Atoms distributed randomly in space and velocity direction –Equal probabilities Atoms have speed distribution –Can be specified

11. Applications of Kinetic Theory You can use kinetic theory to to relate the pressure and volume to the speed of the atoms PV = (Nm/3) v 2 –where m is mass per atom Ideal gas law: PV = nRT nRT = (Nm/3) v 2 v = (3nRT/Nm) ½ = (3RT/M) ½ For a given sample of gas v depends only on the temperature

12. 12 Kinetic Energy Since kinetic energy = ½mv 2, K.E. per particle is: K.E. = (3/2)(R/N A )T = (3/2)kT where N A is Avogadro’s number and k is the Boltzmann constant Since internal energy is the sum of the kinetic energy for all particles U = (3/2)NkT