# Isobaric process

An isobaric process is a thermodynamic process in which the pressure stays constant; [itex] \Delta P = 0 [itex]. The heat transferred to the system does work but also changes the internal energy of the system:

[itex] Q = \Delta E + W \qquad \qquad \qquad (1)[itex]

according to the first law of thermodynamics, where W is work done by the system, E is internal energy, and Q is heat. Pressure-volume work (by the system) is defined as

[itex] W = P \Delta V [itex]

but since pressure is constant, this means that

[itex] W = \Delta (P V) \qquad \qquad \qquad (2) [itex].

Applying the ideal gas law, equation (2) becomes

[itex] W = n R \Delta T \qquad \qquad \qquad (3) [itex]

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

[itex] \Delta E = n C_V \Delta T \qquad \qquad \qquad (4) [itex]

then substituting equations (3) and (4) into equation (1) produces:

[itex] Q = n C_V \Delta T + n R \Delta T [itex]
[itex] = n (C_V + R) \Delta T [itex].

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

[itex] C_P = C_V + R [itex]

and if the gas involved in the isobaric process is monatomic then CV = (3/2)R and CP = (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 [itex] Q = \Delta E [itex]. It would be convenient to have a similar equation for isobaric processes. Substituting equation (2) into equation (1) yields

[itex] Q = \Delta E + \Delta (P V) = \Delta (E + P V). [itex]

The quantity E + 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

[itex] Q = \Delta H [itex].

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