Answer each of the following questions. Your answer should fit in the allotted space.

$1.$ Two copper blocks $($a hot block and a cold block$)$ are placed in contact until their temperatures are the same. Throughout the process, the blocks are isolated from the rest of the universe.

a$.(4$ points$)$ As a result of this process, does the entropy of the hot block increase, decrease, or remain the same? Explain your answer using the definition of dS$.$

b$.(6$ points$)$ Is your answer to part a$.$ consistent with the second law of thermodynamics? Provide a complete and convincing explanation of why or why not by considering the entropy change of the universe.

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$2.(16$ points$)$ Consider the van der Waals equation of state. Develop equations for the van der Waals constants in terms of the critical temperature, Tc and critical pressure, Pc$.$ Show your work below.

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$3.$ An engineer claims to have invented steady flow device which takes a $2\mathrm{.}0$ mol$/$min inlet air stream at $3\mathrm{.}4$ atm and $63$oC and exhausts two streams of equal mass. She claims that one of the outlet streams has a temperature of $23$oC and a pressure of $3\mathrm{.}2$ atm, while the other has a temperature of $83$oC and a pressure of $3\mathrm{.}2$ atm. The device neither requires nor produces work, however it releases heat to the surroundings at a rate of $582$ J$/$min$.$ The surroundings maintain a constant temperature of $23$oC$.$

Assume air is an ideal gas with Cp $=3\mathrm{.}5$R where R is the gas constant $(8\mathrm{.}314$ J mol$-1$ K$-1)$

a$.(10$ points$)$ Does this process violate the first law of thermodynamics? Perform the energy balance and show all your work below.

b$.(10$ points$)$ Does this process violate the second law of thermodynamics? Again, show all your work below and on the next page.

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Extra space for problem $3$ b$.$

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$4.(20$ points$)$ Express the following derivative in terms of P$,$ T$,$ V$,$ Cp$,$ Cv$,$ and their derivatives.

$($V$/$P$)\_$H

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$5.$ a$.(10$ points$)$ Consider one mole of a gas which obeys the following equation of state:

P$=$ RT$/$V$-$ a$/$V$^2$ Here $$a$$ is a positive constant.

Derive an expression for the work associated with a reversible, isothermal expansion from an initial volume of V$1$ to a final volume of V$2.$

b$.(10$ points$)$ Consider a gas that obeys the following equation of state:

P$($V$-$b$)=$ RT here $$b$$ is a positive constant.

The heat capacity of the gas depends upon temperature according to the following equation:

Cp $=$ a $+$ cT here $$a$$ and $$c$$ are positive constants.

Derive an expression for the change in entropy of one mole of this gas associated with a change in state from P$1$ and T$1$ to P$2$ and T$2.$

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Extra space for $5$ b$.$

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$6.(14$ points$)$ Consider the T$-$S diagram shown below for a cyclic process that consists of two isothermal step $($an expansion for b to c and a compression from d to a$)$ and two adiabatic steps $($an expansion from c to d and a compression from a to b$).$ Note that point b$$ has the same entropy as point b and point c$$ has the same entropy as point c$.$

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Write the appropriate area $$A$$ through $$G$$ beside each of the descriptions below:

The maximum work that can be produced by operating this cycle with the isothermal expansion at T$2$ and the isothermal compression at T$1.$

The heat that is unavailable for work due to the isothermal compression being at T$1$ instead of T$0.$

The heat that is unavailable for work due to the isothermal compression taking place at T$0$ instead of absolute zero.

The heat that is unavailable for work due to the irreversibility of expansion c$-$d and due to the isothermal compression taking place at T$0$ instead of absolute zero.

The heat that is unavailable for work due to the irreversibility of expansion c$-$d and due to the isothermal compression taking place at T$1$ instead of T$0.$

The heat that is unavailable for work due to the irreversibility of compression a$-$b and due to the isothermal compression taking place at T$1$ instead of T$0.$

The heat that is unavailable for work due to the irreversibility of compression a$-$b and due to the isothermal compression taking place at T$0$ instea