An ideal gas with constant heat capacities and (so that and ) is held in a lagged…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics · The laws of thermodynamics

Thermal Physics, Ch The first law: work, heat and internal energy

Problem

An ideal gas with constant heat capacities

C_V

and

C_p

(so that

C_p - C_V = Nk_B

and

\gamma = C_p/C_V

) is held in a lagged cylinder by a light frictionless piston, so that no heat enters or leaves the gas. The piston is moved slowly, so the change is quasistatic and

\delta W = -p\,\mathrm{d}V

. ((a)) Starting from the first law

\mathrm{d}U = \delta Q + \delta W

, with

\mathrm{d}U = C_V\,\mathrm{d}T

, and eliminating

T

with

pV = Nk_BT

, show that along such a change

\dfrac{\mathrm{d}p}{p} = -\gamma\,\dfrac{\mathrm{d}V}{V}

, and hence that

pV^{\gamma} = \text{const}

. ((b)) Helium, a monatomic gas with

\gamma = 5/3

, is at

1.0\times10^5\ \mathrm{Pa}

and is compressed in this way to half its volume. Find its final pressure.

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