Page 32 - Basic physical chemistry for the atmospheric sciences

1 8 Basic physical chemistry

dq - d w = du (2.2)
where dq is the differential increment of heat added to a unit mass of
the system, dw the differential increment of work done by a unit mass
of the system, and du the differential increment in internal energy of a
unit mass of the system. Equations (2. 1 ) and (2.2) are statements of
the.first law o f thermodynamics. In fact , Eq. (2. 2 ) provides a definition
of du. It should be noted that the change in internal energy du is a
function only of the initial and final states of the system and is there­
fore independent of the manner by which the system is transferred
between these two states. Thermodynamic variables that possess this
property are called f u nctions o f state. For example, pressure volume
and temperature are functions of state.
To visualize the work term dw in Eq. (2.2) in a simple case, consider
a substance (often called the working substance) contained in a cylin­
der of fixed cross-sectional area which is fitted with a movable, fric­
(
tionless piston F ig. 2. 1 ) . The volume of the substance is then propor­
tional to the distance from the base of the cylinder to the face of the
piston, and can be represented on the horizontal line of the graph
2
shown in Figure . 1 . The pressure of the substance in the cylinder can
be represented on the vertical line of this graph. Therefore, every state
of the substance corresponding to a given position of the cylinder is
represented by a point of the graph. When the substance is in equilib­
P
rium at a state represented by the point o n this graph, its pressure is
p and its volume V. If the piston moves outward through an incremen­
tal distance dx, while the pressure remains essentially constant at p ,
b
the work dW done y the substance in expanding is equal o the force
t
exerted on the piston (this force is equal to pA where A is the cross­
sectional area of the piston) multiplied by the distance dx through
which the piston move . That is,
s
d W = pA dx =p dV (2.3)

d
In other wor s , the work done by the substance when its volume
increases by a small amount is equal to the pressure of the substance
i
multiplied by its n crease in volume. It should be noted that dW = p dV
is equal to the shaded area in the graph shown in Figure 2. 1 ; that is, it
is equal to the area under the curve PQ. When the substance passes
from state A with volume V1 to state B with volume V2 (Fig. 2 . 1 ) ,
during which its pressure p changes , the work W done b y the sub­
stance is equal to the area under the curve AB. That is,