MATSE 201 (Introduction to Materials Science and Engineering)
HW4: Thermodynamics (100 pts.)
Dawson Alexander
For full credit clearly state your approach, assumptions, and calculation steps. Use units where
needed and clearly mark your answer. Upload your solutions as a .pdf file to Canvas.
1. Laws of Thermodynamics
a. Write the 1st law of thermodynamics in differential form for a closed system. (6 pts.)
𝒅𝑼 = 𝒅𝑸 + 𝒅𝑾
(𝟏)
b. If a system gains heat from the surroundings, is dQ positive or negative? If a system does
work on the surroundings, is dW positive or negative? (4 pts.)
If a system gains heat from the surroundings, then dQ is positive.
If a system does work on the surroundings, then dW is negative.
c. Write an expression for the 2nd law of thermodynamics (either classical or statistical
mechanics). (6 pts.)
𝒅𝑺𝒔𝒚𝒔 + 𝒅𝑺𝒔𝒖𝒓 > 𝟎
(𝟐)
d. The “Heat Death of the Universe” is a potential fate of the universe. Explain how this
hypothesis is arrived at based on the 2nd law of thermodynamics as well as what type of
system it assumes the universe to be. (14 pts.)
The universe is assumed to be an isolated system. If it is an isolated system then there is a
limited amount of energy within the universe. Whenever you have energy transferred in an isolated
system the total amount of entropy in the systems increases. Entropy never decreases, it will always
increase. Since entropy always increases this means that theoretically the universe as we know it will
reach a point where it is at maximum entropy. Since the conversion of heat and work into entropy is an
irreversible process when the universe reaches its maximum entropy, the universe will eventually be a
cold, dark, and boring.
2. Entropy
a. Define the term “entropy”. (6 pts.)
A thermodynamic property related to the degree of randomness or disorder in a system. It
also could be considered a measure of the degradation of energy quality in a system.
b. Rank the following systems from least to most entropy, assuming the same temperature:
liquid, gas, solid. (12 pts.)
1. Gas – Most Entropy
2. Liquid
3. Solid – Least Entropy
c. Assume that you have a system consisting of three boxes (A, B, and C) and two particles
(R and B) that can be placed into the boxes (each box can hold both particles, one, or none).
i. Draw all distinct microstates for this system. (10 pts.)
The total number of microstates within the system is 16 when it is assumed that the system can also exist
when the one or both particles are absent from the system.
Figueres 1-16: Microstates in the system.
ii. Calculate the entropy of this system. (10 pts.)
The entropy of a system can be determined by calculated by
𝑺 = 𝒌𝑩 𝐥𝐧(𝜴)
(𝟑)
-5
where kB is the Boltzmann (1.381*10 J/K or 8.617*10 eV/K) constant and the Ω is the number
of microstates within the system. Ω = 16.
S = 3.83*10^-23 J/K
-23
3. State Functions
a. Define the term “state function”. (6 pts.)
A state function is a property whose value is independent of path.
b. Of the following, identify which are state functions and which are path functions: “heat”,
“elevation”, “volume”, “temperature”, “work”, “distance travelled”, “pressure”, “internal
energy”, “Gibbs free energy”. (10 pts.)
 Heat – Path Function
 Elevation - State Function
 Volume – State Function
 Temperature – State Function
 Work – Path Function
 Distance Traveled – Path Function
 Pressure – State Function
 Internal Energy (U) – State Function
 Gibbs Free Energy - State Function
c. Explain how the Gibbs free energy can still be negative (spontaneous) for an endothermic
(+ΔH) reaction. (10 pts.)
𝜟𝑮 = 𝜟𝑯 − 𝑻 ∗ 𝜟𝑺
(𝟒)
If the reaction is endothermic then you can still have a reaction that is spontaneous because if the
change of entropy is sufficiently positive or the temperature is high enough then the product of the
temperature and the change in entropy can offset the change in enthalpy and result in a negative value for
Gibbs Free energy.
d. Define the term “free energy”. (6 pts.)
In thermodynamics free energy is the energy available in a system to do work (useful energy).