Page 53 - Geothermal Energy Renewable Energy and The Environment

36 Geothermal Energy: Renewable Energy and the Environment

T = T i
V = V i
P = P i

T = T i
1 V = V 1
4 P = P 1
P
T = T 2 2
V = V 3
P = P 3 3 T = T 2
V = V
P = P 2 2

V i V 3 V 1 V 2
V

FIGUre 3.3 Pressure versus volume graph for the series of changes for the Carnot cycle depicted in Figure 3.2.

temperature. If these two materials were placed in contact, the warmer would cool down by transfer-
ring heat to the cooler material and the cooler would warm up by increasing its heat content—the uni-
versal observation that heat always flows spontaneously from a warmer to a colder material. The end
temperature for both materials would be the same but their respective heat contents had to be different.
Together, these observations lead to an important conclusion. Namely, that different materials will
reach a state of equilibrium (that is, a state at which no heat flows between them) only when they are at
the same temperature. And yet, the amount of heat each material contains will not be the same.
These conclusions lead to the concept that materials, for some physical reason that was unknown
at the time, must have specific and unique internal characteristics that determine how much heat
must be added to (or removed from) the material in order for it to change its temperature by a speci-
fied amount. The quantity that represented this phenomenon ultimately became known as the heat
capacity, C, which is expressed in joules per gram for each degree of temperature change (J/g-K).
Eventually, it was recognized that changes in volume and changes in pressure affected the heat
capacity, so it became standard to specify heat capacity either at constant pressure (C ) or constant
p
volume (C ). The mathematical expression for the general relationship for heat capacity and heat is
v
C = dq/dT, (3.9)

where dq and dT are the differential changes in heat and temperature, respectively. Rearranging
Equation 3.9, we see that the amount of heat that can be taken from a system to do work is equal to
the number of degrees the temperature changes multiplied by the heat capacity:

C × dT = dq. (3.10)

As noted previously, for the general case the change in internal energy is Equation 3.5.
From Equations 3.5, 3.6, and 3.10 it follows that, at constant pressure,
dH = C × dT (3.11)
P

and, at constant volume
dE = C × dT, (3.12)
V