This is a closed book, closed notes, quiz. You may only use the “crib sheet” provided below.

You may use a simple (non-programmable, non-graphing) calculator. If you absolutely need to   use a graphing calculator, all memory must be erased. The use of any communication devices   (cell phones, messaging devices, etc.) during examinations is not allowed. Define all symbols  and justify all mathematical expressions used. Make sure to state all of the assumptions used to solve a problem. Credit will be given only for a logical and complete solution that is clearly

communicated with correct units. Partial credit will be given for a well-communicated problem- solving strategy based on correct physics.

Useful Equations and numerical values:

kB  = 1.38 × 10–23  J/K = 8.6 × 10–5  eV/K                                  R = 8.31 J / (mole K)

1 atmosphere (atm) = 1.01 × 105  Pa,   1 Pa = 1 N/m2

1 mole of ideal gas at 1 atm and T = 25 oC occupies a volume of 24.5 liters = 2.45 × 10-2  m3 Water: Latent heat of fusion = 334 Joule/gm, latent heat of vaporization = 2260 Joule/gm

Specific heat of liquid water = 4.18 Joule/(gm- °C)

Specific heat of solid water (ice) = 2.04 Joule/(gm- °C)

Specific heat of water vapor at constant pressure = 1.93 Joule/(gm- °C)

Specific heat of water vapor at constant volume = 1.46 Joule/(gm- °C)

Adiabatic exponent: Y = Cp / Cv = (f + 2)/f; f is the number of degrees of freedom

Adiabatic expansion of an ideal gas: PvY  = constant

One atomic mass unit (amu) = 1:66 x 10–27  kg;

Combinatorics: (|（m(N) =       Stirling’s Approximation: ln N! = N ln N – N

(q + N - 1)

Einstein Solid with N oscillators and total energy U = q ε , Ω = |              |

「   V     3     4πmU    5 ]

For a monoatomic ideal gas:  S = Nk |Lln N + 2 ln  3Nh2     + 2」|

dU = TdS - PdV + μdN ,        dS 之δQT ,                 dS = dT   at constant pressure

dU = δQ - PdV +δWother

Efficiency of an engine operating between a Hot and Cold reservoir = Work out/QH.

Coefficient of Performance of a refrigerator between a Hot and Cold reservoir = QC/Work in

Thermodynamic potentials:  Enthalpy: H = U + PV,  Helmholtz free energy: F = U – TS, Gibbs free energy: G = U – TS + PV

Partition function:《  〉E = −  = kT            , 〈P〉=             

Maxwell speed distribution:   p (v ) = (|（ 2πk(m)BT 32 4πv2 exp  − 2kB(mv)T(2) 

Single-particle partition function for a monatomic ideal gas in box of volume ，:  Z1  = VvQ   ,

where vQ  = 3

Some useful math:

I (n, a ) =  e− ax2 xn dx

I (0, a ) =   a − 12               I (1, a ) =  a − 1

I (2, a ) =    a −32               I (3, a ) =  a −2

I (4, a ) =   a −52               I (5, a ) = a −3

for x < 1, e± x  = 1± x

for x < 1, ln (1± x) = ±x 

Problem 4

The following reaction takes place in our body:

C6 H12O6  + 6 O2  → 6 CO2  + 6 H2O.

In essence, the body oxidizes glucose (C6 H12O6) to generate energy, which in the end allows our muscles to work.

The values for enthalpy, Gibbs free energy, and entropy per mole at 298 K for the interacting substances are given below.

(a) Calculate the change in Gibbs free energy for this reaction

(b) Does this change correspond to energy that is available to do work, or to energy that needs to be provided (by an external source) to make the reaction happen?

(c) Assuming a reversible process, how much energy in the form of heating is expelled to or absorbed from the environment when this reaction takes place? Be sure to specify whether it is absorbed or expelled.

(d) From the definition of Gibbs free energy derive an expression for how ΔG changes with temperature when P and N are fixed and use it to estimate ΔG at 0oC for glucose. You  may assume that the entropy S and enthalpy H are temperature independent.

Multiple Choice Questions (5 choices each, mark the correct answer):

1) Which of the following is always true for the isothermal expansion of an ideal gas?

(a) Q = 0

(b) W = 0

(c) ΔU = 0

(d) Q = W

(e) ΔU = W

2) In equilibrium, coexistence of two phases at a given temperature and pressure requires that

(a) Their energies U are equal

(b) Their entropies S are equal

(c) Their number of atoms N are equal

(d) Their chemical potentials μ are equal

(e) Their enthalpies H are equal

3) An ideal gas occupies one third of a box of thermally insulated volume v. The other two thirds are evacuated. The partition between the two parts is suddenly removed, so that the gas rapidly fills the whole box. Which of the following statements is true?

(a) The gas did work in this process.

(b) The temperature of the gas decreased.

(c) The entropy of the gas remained constant

(d) The entropy of the gas increased

(e) The heat flow into the gas balanced the work done by the gas.

4) Which of the following expressions is NOT a valid Maxwell Relation?

(a)  − (∂S∂P)T  = (∂V∂T )P

(b)  (∂T∂P)S   = (∂V∂S )P

(c)  (∂S∂ V )T  = (∂P∂T )V

(d)  (∂P∂S )V  = − (∂T∂ V )S

(e)  (∂S∂T )V  = (∂P∂ V )T

5) The Clausius-Clapeyron equation describes how the pressure of a phase transition such as the melting of a solid changes as a function of temperature

dPdT = ∆s∆v = (sl  − ss )(vl  − vs ) ,

where sl   and  ss    is the entropy per mole of the liquid and solid phases respectively.

While the melting temperature of many solids increases with increasing pressure, requiring

more energy to melt than without the additional pressure, the melting point of water is lowered by external pressure. This is because

(a)  The entropy per mole of the solid is greater than the entropy per mole of the liquid (b)  The entropy per mole of the solid is equal to the entropy per mole of the liquid

(c)  The volume per mole of the solid is greater than the volume per mole of the liquid (d)  The volume per mole of the solid is less than the volume per mole of the liquid

(e)  None of these are correct

6) What are the natural independent variables of the enthalpy H?

(a) P and V

(b) V and S

(c) P and T

(d) V and U

(e) P and S

7) Evaluation of N ,V   yields

(a) U               (b) V               (c) β                (d) H               (e) μ

8) Imagine that you have 3 different systems, and each system has only two possible states. The relative energy spacing between the states is shown below.

System 1 is at a temperature that is lower than systems 2 and 3, which are at the same temperature. In the figure T：< T₂. Which of the following statements is true?

(a) The partition function of system 1 is larger than that of 2; the probability to be in an excited state is larger in system 2 compared to 1.

(b) The partition function of system 2 is smaller than that of system 3; the probability to be in an excited state is larger in system 2 compared to 3.

(c) It is equally probably to be in the ground state of system 1 and system 2.

(d) The partition function of system 1 is smaller than that of 2; the probability to be in an excited state is larger in 2 than in 3.

(e) none of the above