Physical Chemistry Mock Midterm Exam

Physical Chemistry Mock Midterm Exam

**disclaimer: I have no idea exactly what will be on the midterm exam beyond what was listed by the professor in class, although I do know these section headings are accurate. This mock midterm is written only as a study tool. It’s usefulness is debatable.

Part I: Definitions (20%)

Provide definitions for each of the following terms. Do NOT give any examples of the terms or you will lose marks.  (1 mark per definition, -1 mark if example given)

1) System:

2) Isolated system:

3) closed system:

4) open system:

5) surroundings:

6) Zeroth Law of Thermodynamics:

7) Thermal Energy:

8) Internal Energy:

9) Heat:

10) Work:

11) First Law of Thermodynamics:

12) Reversible Path:

13) Nonreversible Path:

14) State Function:

15) Heat Capacity (also include definition for molar heat capacity and specific heat capacity):

16) Translational State (strongly doubt this would actually be asked because not defined in lecture):

17) Rotational State:

18) Vibrational State:

19) Equiparition of Energy:

20) Bond Energy

21) Hess’s Law:

22) Heat of Formation of a Molecule:

22) Heat of Combustion of a Molecule:

23) Enthalpy:

26) State: 

Part II: Calculations

Provide all work, including appropriate equations, notation, and manipulation.

Try (or retry) calculations from the textbook without looking at the solutions manual.

For a good selection of questions do:

Energy, Heat, Work

3, 4, 6, 10

Thermochemistry

15, 17, 21, 24, 29, 32, 34

Equiparitioning of Energy

1)   Calculate the number of degrees of freedom for carbon dioxide. Estimate the molar heat capacity of the substance.

2)   Calculate the number of degrees of freedom for methane. Estimate the molar heat capacity of the substance.

Part III: Short Answers

1)   What type of system is a coffee cup calorimeter? What type of system is a bomb calorimeter? Compare and contrast.

2)   If system A is 25ºC and system B is 25ºC, and system C is in equilibrium with system B, what can we conclude about the temperature of system C and its relationship with system A?

3)   Draw potential energy diagram of a diatomic molecule. Label the place of greatest repulsion, A. Label the place of greatest stability, B. Label zero point, C. Label and show the bond energy. Draw in the largest state (rotational/vibrational/translational) , and discuss the placement and magnitude of the other states (rotational/vibrational/translational).

4)   Is the maximum work done by the system reversible or irreversible? Explain referring to a P versus V diagram.

5)   Using the definition of enthalpy, show that it is equal to the heat at constant pressure.

6)   Define both the heat capacity at constant volume and the heat capacity at constant pressure.

7)   Is work a state function? Explain.

8)   Discuss how differential calorimetry finds the unknown heat capacity of a substance.

9)   What is the standard enthalpy of formation of oxygen gas?

10) Why is the bond energy always positive?

11) Hydrogen gas only has 2 rotational degrees of freedom. Explain why.

12) Discuss the effects of temperature when comparing calculated heat capacities using equiparitioning of energy and the actual measured heat capacity values.

Part IV: True or False

This section will NOT be on the exam, however, it may provide a good snapshot as to whether the definitions are understood.

1)   Heat leaves the system. q is positive.             True/False

2)   Energy is stored in chemical bonds.             True/false 

3)   The system does work. w is negative.             True/false

4)   Internal energy does not depend on PV work.             True/false

5)   Bond energy is usually greater than thermal energy.             True/false

6)   A diatomic gas has 2 degrees of translational energy.             True/false

7)   PV work is positive for an expansion of a gas.             True/false 

8)   The lowest vibrational state is at the zero point.             True/false 

9)   The heat capacity at a phase change is infinite.            True/false

10)  Vibrational states are the same as angular momentum.             True/false 

Part V: Bonus Questions

1)   Show that PV=nRT is a state function using partial derivatives and Euler’s Theorem for Exactness without referring to Appendix C. (Just kidding! Bad joke, I know.) (Yes, I am actually joking about this one, don't try to solve it!)

2)   Describe your experience and opinion of the textbook with regards to whether you have the e-book or hardcopy, its acquisition and availability, its use as a study tool, and to what extent it corresponds to and complements the lectures.

Equiparitioning of Energy

This post is designed to explain how to calculate the various degrees of freedom of translational, rotational, and vibrational energy in a molecule, and and calculate heat capacity from these values with equiparitioning of energy.

Let's start by defining internal energy of a molecule:

The bond energy usually dominates the thermal energy with a much bigger value, so we often neglect the thermal energy. However, heat capacity depends only on thermal energy, NOT bond energy, so it is important for something.

Translational Energy

The translational energy is defined as 

I will offer an explanation as to where this definition came from later, but until I update this post, I suggest checking both http://www.chem.queensu.ca/people/faculty/mombourquette/firstyrchem/GasLaws/KMT.htm

and section 1.9 from the text book. (My laziness is because I am incredibly hungry and tired.) 

The translational energy is partitioned among three degrees of freedom along the three cartesian axes. 

Rotational Energy

The rotational energy also is split up equally between three degrees of freedom. However, with a linear molecule, rotating it about the z-axis has insignificant energy as the mass of the molecule is almost completely in the atomic nuclei, so we only consider its rotation about the x- and y-axis.

A linear molecule has only two degrees of rotational freedom. 

A nonlinear molecule has three degrees of rotational freedom. 

Vibrational Energy

The vibrational energy can be resolved into a certain number of normal modes of vibration, known as its degrees of freedom. 

Summary of Degrees of Freedom

If a molecule has N atoms, 3N degrees of freedom must be specified. We have already defined the translational and rotational degrees of freedom for nonlinear and linear molecules. The vibrational degrees of freedom are merely the leftovers from the total. This is summarized in the table below: 

Heat Capacities Further Defined

Equiparition of energy states that each translational and rotational degree of freedom contains 1/2RT energy, of kinetic energy only. Each vibrational degree of freedom contains RT energy, which includes both the potential energy and kinetic energy. 

The heat capacities are given by multiplying the energy factor by the gas constant times the number of degrees of freedom. This gives a value in J/(mol*K), which is the unit for heat capacity. 

These formulae can be used to estimate the molar heat capacities of gases. The molar heat capacity of monatomic gases only depends on translational motion. For diatomic molecules, the translational and rotational components are more dominant than the vibrational components for determining the heat capacities. For anything larger, the calculated values are very far off the actual heat capicities. So instead of throwing our hands up in the air and praying to the quantum mechanic gods, mortals immortalized (Einstein and Stern) suggest that this extreme variation from the actual values is because equiparitioning breaks down at lower temperatures, especially for vibrational states. 

Why is that? Remember that vibrational levels are much more widely spaced than the rotational and translational levels. The larger gaps require a large quantum of energy to be supplied to move from one vibrational level to the next. This would sufficiently muck up our calculated values. However, at higher temperatures the calculated molar heat capacity values would have much better agreement with actual values, as more energy is supplied to the molecules. 

Summary: 

Molecules have various degrees of freedom which compose their thermal energy. These degrees of freedom are related to the heat capacity. By knowing the number of degrees of freedom and how energy is distributed between the degrees of freedom (translational, rotational and vibrational), we then can construct the heat capacities of the substance. 

At lower temperatures, the heat capacities calculated in this manner will not correspond with the actual heat capacities.

Because two hot plates are better than one...

How my TA instructed us to set up our distillation for the extraction of trimyristin from nutmeg lab.

Note the beaker set up to catch the water dripping from a leak in the tubing. 

Translational, Rotational, and Vibrational Energies

Energy from a molecule is not stored in chemical bonds. Rather it is stored in the translational, rotational, and vibrational energies of the molecule - the thermal energy.

The Potential Energy Diagram for a Diatomic Molecule

The states of a molecule are quantized and can be represented by a potential energy diagram. This potential energy diagram represents a diatomic molecule. On the x axis, we examine the internuclear distances between the two atoms. As the atoms approach an internuclear distance of zero, their potential energy is infinite. This is because the two atoms cannot occupy exactly the same point at the same time. Repulsion forces dominate, and there is no chemical bond. The point with the lowest potential energy of the graph is the ideal bond length of the diatomic molecule. Then, for increasing values of internuclear distance between the nuclei of the two atoms the potential energy increases and no bonding occurs as the attractive forces are too weak. Essentially, the two atoms are too far apart to have a bond. Note that when two atoms approach one another from infinite distance (righthand side of graph), the energy of the system is lowered.

This can be compared to you and a not so close friend sitting on a chesterfield (sofa, couch, whatever) together. Let's call your friend Matthew. If Matthew and you both sit on the far ends of the chesterfield, chances are you are less likely to have a conversation and be friendly together. Other elements in your environment aka other friends could come and sit between you both, and so you might be closer these other friends, and hence more engaged with them. This is analogous to no bonding at great distances. Now if you decide to sit on top of Matthew, he will not be very happy with you because he doesn't know you very well. Chances are he'll go all cactus mode and repel the physical contact. Sounds like what happens to two atoms with a internuclear distance approaching zero! However, to achieve the optimal comfort level, you would sit beside Matthew close enough to be friendly, but far enough away to respect both of your personal bubbles. Hence, you experiencing bonding at this distance.

Now the funky stripes on the potential energy diagram represent the vibrational energy of the molecule. The lowest vibrational state is above the minimum potential energy on the curve. Why? Well, according to the Heisenburg Uncertainty Principle, the both the bond length at equilibrium and the vibrational motion can not simultaneously be known. This point is called the zero point.

Taking a Closer Look

Let's zoom in on some of these vibrational states on the potential energy diagram.

We find in between them superimposable rotational states that also describe the state of the molecule. Even further in between these rotational states we find the translational states of the molecule. This is because

Now, I can't really use the Matthew analogy here, because a vibrating Matthew is just a very, very bad idea (yikes, did I actually just write that?), but I will attempt to explain in greater detail where on earth these states come from!

Molecular Quantum States

The states of a molecule are quantized. Here is how these various quantizations are defined (I realize the definitions here may be crude, however this is the level we are expected to understand for this course, so it isn't too elaborate.):

a) electronic states: recall the s, p, d, f orbitals for atoms. There is a similar method out there for molecular states.

b) rotational states: these the same as other angular momentum, like the angular momentum in orbitals. These are usually denoted by quantum numbers Jmj where J goes from 0 to infinity and -J < mj < +J . Spin also has angular momentum given by either +1/2 or -1/2.

c) vibrational states: go in integer values from v=0 to infinity.

d) translational states: can be eliminated by sitting on the frame reference of the molecule and hence are usually not of interest. This can be done by taking a frame of reference. For insistence if you bus from your apartment to St. Viateur bagel, the motion of the bus is counted as stationary if you consider the frame of reference of the bus. So if you "sit" on top of a molecule, its translational states can be ignored.

The states we examine in thermodynamics are all from the world of the macroscopic.
These states (electronic, rotational, vibrational, translational) discussed above are from the quantum world - the microscopic. However, since the lectures took a substantial aside in this direction to explain heat capacity as it is related to the thermal energy of the substance, the information above is probably relevant.

Chapter 2 Definitions for Chem 203

So our secret (or not so secret rather) instructions say that we need to write out a bunch of definitions for 20% of our midterm, and no examples can be given otherwise marks will be docked.

I'm going to include key terms from each of the lectures here. This list is most likely incomplete, but  I'm going to post what I have so far and add to it later. :) The main sources for these definitions were the lecture notes and the hardcopy of the textbook. And of course, wikipedia. cough, cough.

Lecture 1

system: that part of the universe of interest
open system: a system where matter or energy can flow into/out of the system
closed system: a system where energy can flow into/out of the system, but matter cannot
isolated system: neither matter nor energy can enter/exit the system, only move around inside

temperature: physical property related to the measure of the average kinetic energy of the atoms or molecules of a substance; degree of hotness of a body, substance, medium
thermal equilibrium: any flow of heat into or out of a body is equal to the flow of heat into or out of another body, the two bodies have the same temperature (also applies to more than two bodies)

exothermic: energy is released from the system, weaker bonds go to stronger bonds
endothermic: energy is absorbed by the system, stronger bonds go to weaker bonds

Zeroth Law of Thermodynamics: If system A is in equilibrium with system B and system B is in equilibrium with system C, then system A is in equilibrium with system C and A, B, and C have the same temperature.

state: defined by the parameters that are measured and describe a system under these conditions

Lecture 2 and 3

thermal energy: (this is really difficult to define, so the definition I provide is half-a**ed aka ambiguous from the prof's notes) the energy around us, part of the total internal energy of a thermodynamic system or sample of matter, partly the kinetic energy of the system's particles, equally partitioned between all degrees of freedom of particles (translational, vibrational, rotational), not a state function or property of a system as depends on path taken

*potential energy diagram*
I will discuss the BE diagram of a diatomic molecule in another post, as this shows up time after time and it might be important for the course. :) Yay!

cold temperatures: temperatures less than 1000-2000 K, restricts chemistry

internal energy: energy needed to create system from atoms, contains kinetic energy of motion (moving, rotating, vibrating), as well as electrostatic energy of the attraction and repulsion of electrons and nuclei [excludes PV work on the surroundings]

heat: transfer of energy from high to low temperature by conduction, radiation, convection; path dependent

work: energy transferred by a force acting on a distance; path dependent

PV work: 

state function: property of a system whose value does not depend on the path taken to reach that specific value, only dependent on initial and final conditions

reversible path: maintains equilibrium between the system and surroundings throughout the change from the initial state to the final state; most efficient path

non-reversible path: does not maintain equilibrium between the system and surroundings throughout the change from the initial state to the final state, cannot not be undone, inefficient

The First Law of Thermodynamics: U=q+w

Thermodynamics: gives relationships between states at equilibrium

Lecture 4

heat capacity: the amount of heat required to raise the temperature of any substance by 1 K; includes the ability for a substance to store energy the various degrees of freedom in movement/translation, rotation, and vibration of the particles

Enthalpy: H= U + PV

isochoric process: process occurring at constant volume (the internal energy is equal to the heat of the system)

isobaric process: process occurring at constant pressure (the enthalpy is equal to the heat of the system)

isothermal process: process occurring at constant temperature

Lecture 5

No vocab, just a proof dealio about state functions and Euler's criterion for exactness, which will not show up on the exam as involves material beyond the scope of the course.

Lecture 6

thermochemistry: study of the heats of reactions under various conditions

*bond energy diagram shows up again*

Lecture 7

equipartition of energy: in thermal equilibrium, the energy is shared equally among all of its various forms, such as translational, rotational, and vibrational components of a molecule
^
(stay tuned for an upcoming post about this long topic)


Lecture 8

bond energy: energy needed to break a bond, always positive as energy put into the system

Differential Scanning Calorimetry

This is just a quick post on something that was mentioned briefly in lecture. I strongly doubt that any type of calculation based question would be asked on a midterm, but perhaps a theory question could show up.

Differential Scanning Calorimetry is when you compare an unknown sample with a reference sample of known heat capacity at constant pressure. You choose a reference with a well-defined heat capacity over the range of temperatures scanned. Also, you choose a reference that will not undergo a phrase transition, as that will distort the results in regards to the sample. The differential scanning calorimeter is used to measure heat flow in the substances in either exothermic and endothermic reactions. It has the ability to measure phase transition temperatures and enthalpies, which makes it useful to create phase transition diagrams and the like.

Wasn't that terribly informative?

Infinite Heat Capacity?

How much heat is required to change ice at -10ºC into steam at 110ºC for one mole?

This is a favourite question by any thermal dynamics textbook. :)

First of all, we consider what happens at a phase change. At a phase change, the substance is applying all of the heat received into breaking the bonds, and hence, the temperature does not change. However, when the substance accepts heat and is not at a state change, the substance does increase in temperature. The important temperatures to consider, are of of course the melting point and the boiling point. Of water, we know they are 0ºC and 100ºC respectively.

Note that in these equations, one always is evaluating the difference in temperature. This type of calculation does not require conversion from Celsius to Kelvin as the difference will stay the same. (If you are skeptical, try it out! I only mention this as the question came up in lecture and seemed to be shared by many people.)

So returning to the initial question, how much heat is required?

This equation can be shown heuristically by a graph:

Now, examining the heat capacity of water at constant pressure at the phase changes we see a the following graph. This graph is a simplified version of the real relationship of heat capacity while applying heat to a substance, as the heat capacity between the phase change would not be constant. However, we consider it to be constant to simplify matters.

The intriguing part of the above graph is what happens to the heat capacity at the phase transition. The change in temperature is zero at these places. This can be expressed by:

The heat capacity is infinite at a phase change? What does this mean?

Recall, the system will absorb all of the heat given without changing the temperature, and so until the bonds are broken, it will happily accept ALL the heat provided. The heat capacity is infinite at the phase change, as the system has no limit for receiving heat.