Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 70 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 73 / 1 / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 73 / 1 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 70 / 1 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 71 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 71 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 74 / 4 / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 34D / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 38C / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 65 / 1 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 65 / 1 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 65 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 65 / 3 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 66 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 66 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 34D / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 15A / Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 70 / 2 / Solution

{wiki}

= Perfect gas
{synonym}

An ideal gas is a model of a dilute <gas> whose particle centres move according to <classical mechanics> and occupy negligible <volume (thermodynamics)> compared with their container. Treating the particles as <point masses> removes the molecular-size correction; neglecting interactions between separated particles removes the intermolecular contribution to <potential energy>. The usual <kinetic theory> model retains brief <elastic collisions> with the walls and, optionally, with other particles: these transfer <momentum> while conserving the total <kinetic energy> of the colliding bodies. There is no appreciable interaction <energy> stored between <collisions>. These assumptions lead to the <ideal gas law>, $PV=Nk_BT$. https://openstax.org/books/university-physics-volume-2/pages/2-1-molecular-model-of-an-ideal-gas[OpenStax University Physics Volume 2, section 2.1: molecular model and ideal gas law]{ref} https://openstax.org/books/university-physics-volume-2/pages/2-2-pressure-temperature-and-rms-speed[OpenStax University Physics Volume 2, section 2.2: kinetic-model assumptions]{ref}

The model's attraction is its universality: at the same <temperature> and <number density>, different classical <ideal gases> have the same <pressure>, even when their particle <masses> differ. In <ideal gas pressure from wall collisions>, heavier particles move more slowly at a given <temperature>, and the two effects cancel. For example, one <mole> of any <ideal gas> at $300\,\mathrm K$ in $0.025\,\mathrm{m^3}$ has <pressure> approximately $99.774\,\mathrm{kPa}$, as computed under the <ideal gas law>.

The <ideal gas law> describes the translational contribution to <pressure>; it does not specify all of the <internal energy> or the <heat capacity>. A <monatomic gas> has three classical translational <degrees of freedom>, whereas a <gas> of <molecules> can also store <rotational kinetic energy> and <energy> in vibrational <normal modes>. An <equation of state> alone therefore does not determine how much <heat> a chosen <temperature> increase requires. It also does not select a process: <isothermal expansion> and <adiabatic expansion> need different additional constraints. https://openstax.org/books/university-physics-volume-2/pages/2-3-heat-capacity-and-equipartition-of-energy[OpenStax University Physics Volume 2, section 2.3: internal degrees of freedom and heat capacity]{ref}

A <real gas> departs from this approximation when particle size or interactions become important, especially at high <number density> or low <temperature> near a <phase transition>. The <Van der Waals equation> adds such corrections. Independently, at sufficiently low <temperature> or high <number density>, <quantum statistics> can become essential: even particles with negligible interactions need not obey the classical <ideal gas law>. The classical statistical assumption is the <nondegenerate gas> regime, which is distinct from neglecting interactions. https://openstax.org/books/university-physics-volume-2/pages/2-1-molecular-model-of-an-ideal-gas[OpenStax University Physics Volume 2, section 2.1: departures from ideal-gas behavior]{ref} <Statistical Physics by David Tong/3 Quantum Gases>{ref}

Ideal gas

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