Isothermal–isobaric ensemble

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature T {displaystyle T,} and constant pressure P {displaystyle P,} applied. It is also called the N p T {displaystyle NpT} -ensemble, where the number of particles N {displaystyle N,} is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition. The NPT ensemble is also useful for measuring the equation of state of model systems whose virial expansion for pressure cannot be evaluated, or systems near first-order phase transitions. The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature T {displaystyle T,} and constant pressure P {displaystyle P,} applied. It is also called the N p T {displaystyle NpT} -ensemble, where the number of particles N {displaystyle N,} is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition. The NPT ensemble is also useful for measuring the equation of state of model systems whose virial expansion for pressure cannot be evaluated, or systems near first-order phase transitions. The partition function for the N p T {displaystyle NpT} -ensemble can be derived from statistical mechanics by beginning with a system of N {displaystyle N} identical atoms described by a Hamiltonian of the form p 2 / 2 m + U ( r n ) {displaystyle mathbf {p} ^{2}/2m+U(mathbf {r} ^{n})} and contained within a box of volume V = L 3 {displaystyle V=L^{3}} . This system is described by the partition function of the canonical ensemble in 3 dimensions: where Λ = h 2 β / ( 2 π m ) {displaystyle Lambda ={sqrt {h^{2}eta /(2pi m)}}} , the thermal de Broglie wavelength ( β = 1 / k B T {displaystyle eta =1/k_{B}T,} and k B {displaystyle k_{B},} is the Boltzmann constant), and the factor 1 / N ! {displaystyle 1/N!} (which accounts for indistinguishability of particles) both ensure normalization of entropy in the quasi-classical limit. It is convenient to adopt a new set of coordinates defined by s i = L r i {displaystyle mathbf {s} _{i}=Lmathbf {r} _{i}} such that the partition function becomes If this system is then brought into contact with a bath of volume V 0 {displaystyle V_{0}} at constant temperature and pressure containing an ideal gas with total particle number M {displaystyle M} such that M − N ≫ N {displaystyle M-Ngg N} , the partition function of the whole system is simply the product of the partition functions of the subsystems: The integral over the s M − N {displaystyle mathbf {s} ^{M-N}} coordinates is simply 1 {displaystyle 1} . In the limit that V 0 → ∞ {displaystyle V_{0} ightarrow infty } , M → ∞ {displaystyle M ightarrow infty } while ( M − N ) / V 0 = ρ {displaystyle (M-N)/V_{0}= ho } stays constant, a change in volume of the system under study will not change the pressure p {displaystyle p} of the whole system. Taking V / V 0 → 0 {displaystyle V/V_{0} ightarrow 0} allows for the approximation ( V 0 − V ) M − N = V 0 M − N ( 1 − V / V 0 ) M − N ≈ V 0 M − N exp ⁡ ( − ( M − N ) V / V 0 ) {displaystyle (V_{0}-V)^{M-N}=V_{0}^{M-N}(1-V/V_{0})^{M-N}approx V_{0}^{M-N}exp(-(M-N)V/V_{0})} . For an ideal gas, ( M − N ) / V 0 = ρ = β P {displaystyle (M-N)/V_{0}= ho =eta P} gives a relationship between density and pressure. Substituting this into the above expression for the partition function, multiplying by a factor β P {displaystyle eta P} (see below for justification for this step), and integrating over the volume V then gives The partition function for the bath is simply Δ b a t h = V 0 M − N / [ ( M − N ) ! Λ 3 ( M − N ) {displaystyle Delta ^{bath}=V_{0}^{M-N}/[(M-N)!Lambda ^{3(M-N)}} . Separating this term out of the overall expression gives the partition function for the N p T {displaystyle NpT} -ensemble: Using the above definition of Z s y s ( N , V , T ) {displaystyle Z^{sys}(N,V,T)} , the partition function can be rewritten as