Page 82 - Physical Chemistry

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state 1 to state 2. Each state of a thermodynamic system has a definite value of H. Each Section 2.9
change of state has a definite value of H. Calculation of First-Law Quantities
There are two kinds of quantities for a process. The value of a quantity such as
H, which is the change in a state function, is independent of the path of the process
and depends only on the final and the initial states: H H H . The value of a
1
2
quantity such as q or w, which are not changes in state functions, depends on the path
of the process and cannot be found from the final and initial states alone.
We now review calculation of q, w, U, and H for various processes. In this re-
view, we assume that the system is closed and that only P-V work is done.
1. Reversible phase change at constant T and P. A phase change or phase tran-
sition is a process in which at least one new phase appears in a system without the
occurrence of a chemical reaction. Examples include the melting of ice to liquid
water, the transformation from orthorhombic solid sulfur to monoclinic solid sulfur
(Sec. 7.4), and the freezing out of ice from an aqueous solution (Sec. 12.3). For now,
we shall be concerned only with phase transitions involving pure substances.
The heat q is found from the measured latent heat (Sec. 7.2) of the phase
2
change. The work w is found from w PdV P V, where V is calcu-
1
lated from the densities of the two phases. If one phase is a gas, we can use PV
nRT to find its volume (unless the gas is at high density). H for this constant-
pressure process is found from H q q. Finally, U is found from U
P
q w. As an example, the measured (latent) heat of fusion (melting) of H O at
2
0°C and 1 atm is 333 J/g. For the fusion of 1 mol (18.0 g) of ice at this T and P,
q H 6.01 kJ. Thermodynamics cannot furnish us with the values of the
latent heats of phase changes or with heat capacities. These quantities must be
measured. (One can use statistical mechanics to calculate theoretically the heat
capacities of certain systems, as we shall later see.)
2. Constant-pressure heating with no phase change. A constant-pressure process
is mechanically reversible, so
2
w w rev P dV P ¢V const. P
1
where V is found from the densities at the initial and final temperatures or from
PV nRT if the substance is a perfect gas. If the heating (or cooling) is reversible,
then T of the system is well defined and C dq /dT applies. Integration of this
P
P
equation and use of H q give
P
P T 2
¢H q C 1T2 dT const. P (2.79)
P
T 1
Since P is constant, we didn’t bother to indicate that C depends on P as well as
P
on T. The dependence of C and C on pressure is rather weak. Unless one deals
P V
with high pressures, a value of C measured at 1 atm can be used at other pres-
P
sures. U is found from U q w q w.
P
If the constant-pressure heating is irreversible (for example, if during the
heating there is a finite temperature difference between system and surroundings
2
or if temperature gradients exist in the system), the relation H C dT still
1 P
applies, so long as the initial and final states are equilibrium states. This is so be-
cause H is a state function and the value of H is independent of the path
2
(process) used to connect states 1 and 2. If H equals C dT for a reversible
1 P
2
path between states 1 and 2, then H must equal C dT for any irreversible path
1 P
between states 1 and 2. Also, in deriving H q [Eq. (2.46)], we did not assume
P
the heating was reversible, only that P was constant. Thus, Eq. (2.79) holds for
any constant-pressure temperature change in a closed system with P-V work only.