A gas bottle of fixed volume obeys

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians

Problem

A gas bottle of fixed volume obeys p=nRT/Vp = nRT/V, with R=8.314 J mol−1 K−1R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}. At some instant it holds n=2.00 moln = 2.00\ \mathrm{mol} at T=300 KT = 300\ \mathrm{K} in V=0.0500 m3V = 0.0500\ \mathrm{m^3}. ((a)) Find (∂p/∂n)T,V(\partial p/\partial n)_{T,V}, (∂p/∂T)n,V(\partial p/\partial T)_{n,V} and (∂p/∂V)n,T(\partial p/\partial V)_{n,T} at this instant, and say in words what is held fixed in each. ((b)) A slow leak lets 0.020 mol0.020\ \mathrm{mol} escape while the gas warms by 1.5 K1.5\ \mathrm{K}, the volume staying fixed. Use the total differential to estimate the change in pressure.

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