Download Thermodynamics Lecture 9: Expansion Work and the First Law and more Study notes Physical Chemistry in PDF only on Docsity!
Lecture 9 – Chapt 7 (second half) PV work.
Outline: System and Surroundings First Law ∆U = q + w Expansion Work
- Free expansion
- Isothermal, constant P
- Isothermal reversible
- Adiabatic expansion
Quick review:
We will quickly get through to our fundamental thermodynamic equations for S and U.
Begin with definitions S = S (U ,V,N ) U = U ( S,V,N )
We can then easily define dS and dU.
dS
S
U
dU
S
V
dV
S
N
dN V N U N i (^) U V N
i i (^) j i
^
^ +^
^
^ +^
≠
dU
U
S
dS
U
V
dV
U
N
dN V N S N i (^) S V N
i i (^) j i
≠
and we define the coefficients as familiar driving forces
dS T
dU
p T
dV T
idNi i
dU TdS pdV (^) i dNi i
Remember with thermodynamics, we are keeping track of energy as it flows around. While ago I put up these sketches of heat and work from a microscopic perspective. Remember?
We can think of energy as the capacity to do work. Heat is energy that is transferred by a temperature gradient.
Today we will get into several descriptions of energy flow using the first law.
One final point about work and energy before we get rolling is how we define what is positive. The system is our set of particles of interest. In the above example it is the gas inside the piston. The surroundings is everything else. Positive work, then, is work done on the system by the surroundings. This is positive because work in this direction increases the energy of the system.
OK then, so let’s recall the First law of Thermodynamics: The internal energy of an isolated system is constant.
∆ U = q + w
In other words: energy is neither produced or consumed. In yet another way, q and w are the only ways to change energy.
As pointed out by Abby last week, note that q and w are what kinds of functions? path fnctns But their sum is a? state fnctn.
One common kind of work: Expansion work: work arising from a change in volume. This takes place in internal combustion engines. So, why not picture this as a gas expanding against a piston.
Notice that the work is just the area under the P-V curve. We will use this a lot.
3. Isothermal reversible expansion: Reversible change in thermodynamics is a change that can be reversed by an infinitesimal modification of a variable. Essentially we are always at equilibrium
( ∆ S = 0). Thus, Pext must be equal to Pgas throughout the expansion.
We can put together a hypothetical example of a piston with a pile of sand on it. Slowly removing sand from piston:
Our work integral is the same, but Pext (which = Pgas )is not constant. So, we can use any of our favorite gas equations to describe P. We’ll use ideal gas right now, but could be Van der Waals or whatever. w = - I Pext d V Pext = nRT / V
So, w = -I nRT d V/V = - nRT I d V / V ( why can we pull out T? isothermal) = - ( nRT ln Vf - nRT ln Vi ) = - nRT ln( Vf / Vi )
Note that reversible path is the maximum amount of expansion work (minimum compression). Why? Because reversible by definition utilizes the smallest possible pressure to induce change in volume. If we were irreversible, then Pext would be at the final value for the entire expansion ö Pext-irreversible < Pext-reversible at all points except the very end.
Notice also that doubling the volume at low temperature (or with few molecules present) is less work than doubling the volume at high temperature (or with many molecules present). Remember that these items came from Pext – so higher T or n, means that Pext has to be greater to ensure reversibility.
- Reversible adiabatic expansion :
Remember isothermal was ∆ T = 0 For adiabatic what is zero?
heat: q = 0
Adiabatic gives less work than isothermal. We will show in a second that T drops as the volume increases – thus P drops even more than in the isothermal case.
What does the first law let us say about an adiabatic expansion? ∆ U = w
So, then
If Cv is constant ( for what gas? Ideal) must have no vibration and no distance-dependent interactions ö Ideal monatomic gas.
Then w = CV ∆ T
During expansion, work is done on the surroundings ö w < 0 ... ∆ T < 0
This is our proof that adiabatic reversible is less work than isothermal reversible
