Download Lecture 3 - McMaster Physics and Astronomy

Lecture 3 (Schroeder, chapter 1.3).
Heat. Work.
Internal Energy. First Law of Thermodynamics.
Caloric Equation.
Exact and Inexact Differentials.
Differential Form of First Law.
Problems.
Suppose, we have a container full of gas or some other thermodynamic system. There
are all sorts of mechanisms by which energy can be put or taken out of a system. However,
we usually classify these mechanisms under two categories: heat and work.
Heat is defined as any spontaneous flow of energy from one object to another caused
by a difference in temperature between the objects. Work, in thermodynamics, is defined as
any other transfer of energy into or out of a system.
Again, to further clarify matters we should give a precise definition of energy which
is, as we already know, impossible.
Problem (Schroeder 1.26): A battery is connected in series to a resistor, which is
immersed in water. Would you classify the flow of energy from the battery to the resistor as
“heat” or “work”? What about the flow of energy from the resistor to the water?
Answer:
The flow of energy from the battery to the resistor is work.
The flow of energy from the resistor to the water is heat.
Now, we can formulate the first law of thermodynamics:
The internal energy of a thermodynamic system is a state function of the
thermodynamic coordinates U = U ( p,V , T ) and the change in the energy of a system is
the sum of the heat added to it and the work done on it:
∆U = Q + W .
(3.1)
A few important remarks about the first law.
First, the function U = U ( p,V , T ) is assumed to exist, even though its mathematical
form may not be known (remember that there is the same situation with the state equation:
f (P, V , T ) = 0 ). As a result, the change in internal energy ∆U = U 2 − U 1 resulting from
some process depends only on the initial and final states of the system. It does not
depend on the intermediate states through which the system passes in going from state 1 to
state 2.
Second, the important consequence of the equation of the state f (P,V , T ) = 0 that the
only two of p,V,T are independent variables. As a result, we can write the internal energy
function as either U = U (V , T ) or U(p,T) or U(p,V). The form which is most convenient
depends on the circumstances. In any case it is called the caloric equation of the state of a
thermodynamic system. Again, it is impossible to derive this equation from general
principles of thermodynamics. Thermodynamics takes it as granted either from experiment
or from statistical mechanics.
Third, only the difference ∆U = U 2 − U 1 has physical meaning, as in the case of the
potential energy in Newtonian mechanics. Thus, a constant can be added to the function
U ( p,V , T ) without changing its physical significance. In practice, the constant can be chosen
in any way that is convenient.
Fourth, heat is not a property of a system like temperature, pressure, volume or
mass. It is energy in transit – energy that enters or leaves a system as a consequence of a
temperature difference between the system and a body with which it is in thermal contact
(including radiant heat). Neither work is a property of a system. One cannot say, for
example, “there are W joules of work stored in that system.” Work is energy exchanged
between the system and its surroundings due to a change in one or more parameters that
define the configuration of the system (for example, the displacement of a piston).
Fifth, Q is positive for heat added to the system and negative for heat flowing out of
the system. W is positive for work done on a system and negative for work done by the
system.
In studing thermal processes it is improtant to distinguish between exact differentials
and unexact differentials. A differential which is the total differential of some function is
called an exact differential. Otherwise it is an inexact differential.
If we regard the internal energy function as U = U (V , T ) , then
 ∂U 
 ∂U 
dU = 
 dV + 
 dT
 ∂V T
 ∂T V
(3.2)
is an exact differential. Other choices for the independent variables lead to corresponding
expressions for dU. In every case dU is an exact differential since it is the total differential
of the internal energy function.
In contrast, if an infinitesimal amount of heat dQ is added to a system, dQ is an
inexact differential. It is not the total differential of some function since there is no such
thing as “the heat in a system”. In the same way, an infinitesimal amount of work done by a
system is an inexact differential dW, there is no such thing as “the work in a system”. It will
be helpful to indicate inexact differentials using notation δ Q , δ W instead of dQ, dW.
After this discussion, let us write the first law of thermodynamics for an infinitesimal
process as
dU = δ Q + δ W .
(3.3)
Now if we were to intergate this equation over some thermal process which results in a
given system going from state 1 to state 2, we must recover (3.1).
Problem: The height of the open U-tube is 100 cm. The tube contains mercury to the height
of 50 cm. The atmospheric pressure is 75 cm of mercury. Then, the left side of the tube is
closed at the top, and the right side is connected to a good vacuum pump. How far does the
mercury level fall in the left side? The temperature remains constant.
Answer: Till the height of 25 cm.
Problem: Some amount of gas has changed its state from U1 = 600 kj to U2=200kj. The
work done by the gas was W=300kj. How much heat did the gas get in this process?
Answer: Q= -100 kj.
Problem: Caloric equation for some gas: U = a ln(T T0 ) + b ln( p p0 ) , a = 3 kj , b = 7 kj ,
T0 = 200 k , p 0 = 10 kPa . In some process the gas got Q=14.33 kj and did the work W = 4.56
kj. What is the final gas pressure, if the temperature goes up from 250 to 500 k and the
initial pressure was 105 Pa?
Answer: 3 x 105 Pa.