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First Law -- Part 2

Physics 313Professor Lee

CarknerLecture 13

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Ideal Gas• At low pressure all gases approach an ideal

statelim (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 = CVdT +PdV dQ = CPdT -VdP

Heat Capacities• Heat capacities defined as:

CV = (dQ/dT)V = (dU/dT)V

CP = (dQ/dT)P

• Heat capacities are a function of T only for ideal gases:

• Monatomic gascV = (3/2) R cP = (5/2) R

• Diatomic gascV = (5/2) R cP = (7/2) R

= cP/cV

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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

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Adiabatic Relations

dQ = CVdT + PdV

dQ = CPdT -VdP

VdP =CPdT

PdV = -CVdT

(dP/P) = - (dV/V)PV = const.

• Plotted on a PV diagram adibats have a steeper slope than isotherms

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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

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Finding

• The pressure and the volume changes are related to the force and displacement

• Also related to PV and Hooke’s law = (42mV)/(A2P2)

– Where is the period

• Modern method uses a magnetically suspended piston (very low friction)

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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

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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

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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

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) v2

– where m is mass per atom

• Ideal gas law:PV = nRT

nRT = (Nm/3) v2

v = (3nRT/Nm)½ = (3RT/M)½

• For a given sample of gas v depends only on the temperature

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Kinetic Energy

• Since kinetic energy = ½mv2, K.E. per particle is:

K.E. = (3/2)(R/NA)T = (3/2)kT

• where NA 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