Why is change in enthalpy zero for isothermal processes?

Question

Cameron Andrews · Accepted Answer

The CHANGE in enthalpy is zero for isothermal processes consisting of ONLY ideal gases.

For ideal gases, enthalpy is a function of only temperature. Isothermal processes are by definition at constant temperature. Thus, in any isothermal process involving only ideal gases, the change in enthalpy is zero.

The following is a proof that this is true.

From the Maxwell Relation for the enthalpy for a reversible process in a thermodynamically-closed system,

#dH = TdS + VdP#, #" "bb((1))#

where #T#, #S#, #V#, and #P# are temperature, entropy, volume, and pressure, respectively.

If we modify #(1)# by infinitesimally varying the pressure at constant temperature, we get:

#((delH)/(delP))_T = T((delS)/(delcolor(red)(P)))_(color(red)(T)) + Vcancel(((delP)/(delP))_T)^(1)# #" "bb((2))#

Now, examine the entropy term, which changes due to the change in pressure at constant temperature.

The Gibbs' free energy is a function of temperature and pressure from its Maxwell Relation for a reversible process in a thermodynamically-closed system:

#dG = -SdT + VdP# #" "bb((3))#

Since the Gibbs' free energy (as with any thermodynamic function) is a state function, its cross-derivatives are equal

#((delS)/(delP))_T = -((delV)/(delT))_P#, #" "bb((4))#.

Utilizing #(4)# in #(2)#, we get:

#color(green)(bar(|ul(" "((delH)/(delP))_T = -T((delV)/(delT))_P + V" ")|))# #" "bb((5))#

This relation, which is entirely general, describes the variation of the enthalpy due to a change in pressure in an isothermal process.

The ideality assumption comes in when we use the ideal gas law, #bb(PV = nRT)#.

Thus, #V = (nRT)/P#, and #(5)# becomes:

#color(blue)(((delH^"id")/(delP))_T) = -T(del)/(delT)[(nRT)/P]_P + (nRT)/P#

#= -(nRT)/P cancel((d)/(dT)[T]_P)^(1) + (nRT)/P#

#= color(blue)(0)#

Thus, we have shown that for ideal gases at constant temperature, their enthalpy does not change. In other words, we've shown that for ideal gases, the enthalpy is only a function of temperature.

Xavier Avery · Answer

undefined The change in enthalpy ($ \Delta H $) is zero for isothermal processes because the temperature remains constant throughout the process. Since enthalpy is a measure of the heat absorbed or released by a system at constant pressure, and temperature is directly related to the average kinetic energy of the particles in a system, if the temperature remains constant, there is no change in the heat content of the system, resulting in a zero change in enthalpy.