## Description

1 Choose your path wisely

A particular system has the equation of state U = PV +C, where C is an undetermined constant.

1. The system starts at state A, in which P = 0.2MPa and V = 0.01m3. It is taken quasistatically along the path shown in the figure (A → B, B → C, C → A ). Calculate the heat transferred from the surroundings, q, and the work done on the system, w, for each step along the path.

2. Calculate q and w for a quasistatic process starting at A and ending at B along the path P = a + b(V − c)2, where a = 0.1MPa, b = 1 × 103 MPa m−6, and c = 0.02m3.

3. The system exchanges both heat and work with its surroundings along the paths above. An adiabat is a particular quasistatic path along which work is done but no heat is transferred. Find the form of the adiabats P = P(V ) for the system described by U = PV + C. (Hint: Ifdq¯ qs = 0, then dU = ¯dwqs = −PdV . What else does dU equal?)

2 Is it fundamental enough?

The following ten equations are purported to be fundamental equations for various thermodynamic systems. Six, however, are inconsisent with the basic postulates of a fundamental equation and are thus unphysical. For each, plot the relationship between S and U and identify the six that are unacceptable. v0, θ, and R are all positive constants and, in the case of fractional exponents, the real positive root is to be implied.

S = vR02θ!1/3 (NV U)1/3 = θR21/3NUV 2/3

S

S = Rθ 1/2 NU + RθV2 2!1/2 S = Rv203θ! NUV 3

v0

vR0θ3 !1/5 2 2 1/5 = NRln 2UVRθv0

S = 2 N U V S N

S = NRUθ 1/2 exp −2NV22v02! S = NRU θ 1/2 exp−NRθv UV 0

U = NRθV 1 + S exp(−S/NR) U = v 0θ S2 exp(S/NR) v0 NR R V

3 Find your equilibrium

The fundamental equations of both systems A and B are

R2 !1/3 1/3

S = (NV U)

v0θ

The volume and mole number of system A are 9 × 10−6 m3 and 3mol, respectively, and of system B are 4 × 10−6 m3 and 2mol, respectively. First suppose A and B are completely isolated from one another. Plot the total entropy SA + SB as function of UA/(UA + UB), where UA + UB = 80 J. If A and B were connected by a diathermal wall and the pair allowed to come to equilibrium, what would UA and UB be? 4 Exactly right

The Helmholtz energy A is a thermodynamic state function. Show that

∂A ∂A

= −P and = −S implies

∂V T ∂T V ∂V T ∂T V

5 A difference of degree

Determine whether the following five expressions are homogeneous and, if so, what their degree of homogeneity is:

u = x2y + xy2 + 3xyz

√

u = x + y

u

u = e−y/x

2 + 2y3 u

y

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