# Isobaric process

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An isobaric process is a thermodynamic process in which the pressure stays constant: $\Delta p = 0$ The term derives from the Greek isos, "equal," and barus, "heavy." The heat transferred to the system does work but also changes the internal energy of the system:

$Q = \Delta U + W\,$

According to the first law of thermodynamics, where W is work done by the system, U is internal energy, and Q is heat. Pressure-volume work (by the system) is defined as: (Δ means change over the whole process, it doesn't mean differential)

$W = \Delta (p\,V)$

but since pressure is constant, this means that

$W = p \Delta V\,$.

Applying the ideal gas law, this becomes

$W = n\,R\,\Delta T$

assuming that the quantity of gas stays constant (e.g. no phase change during a chemical reaction). Since it is generally true that[citation needed]

$\Delta U = n\,c_V\,\Delta T$

then substituting the last two equations into the first equation produces:

$Q = n\,c_V\,\Delta T + n\,R\,\Delta T$
$= n\,(c_V + R)\,\Delta T$.

The quantity in parentheses is equivalent to the molar specific heat for constant pressure:

$c_p = c_V + R$

and if the gas involved in the isobaric process is monatomic then $c_V = \frac{3}{2}R$ and $c_p = \frac{5}{2}R$.

An isobaric process is shown on a P-V diagram as a straight horizontal line, connecting the initial and final thermostatic states. If the process moves towards the right, then it is an expansion. If the process moves towards the left, then it is a compression.

## Defining Enthalpy

An isochoric process is described by the equation $Q = \Delta U$. It would be convenient to have a similar equation for isobaric processes. Substituting the second equation into the first yields

$Q = \Delta U + \Delta (p\,V) = \Delta (U + p\,V)$

The quantity U + p V is a state function so that it can be given a name. It is called enthalpy, and is denoted as H. Therefore an isobaric process can be more succinctly described as

$Q = \Delta H \,$.

## Variable density viewpoint

A given quantity (mass M) of gas in a changing volume produces a change in density ρ. In this context the ideal gas law is written

R(T ρ) = M P

where T is thermodynamic temperature above absolute zero. When R and M are taken as constant, then pressure P can stay constant as the density-tempertature quadrant (ρ,T ) undergoes a squeeze mapping. It is this context that explains Peter Olver's use of the term isobaric group when referring to the group of squeeze mappings on page 217 of his book Classical Invariant Theory (1999).