Chapter 14, Critical Phenomena; Higher-Order Phase Transitions Video Solutions, Heat and Thermodynamics

Problem 1

It was shown in Eq. (13.8) that the partition function of a crystal is given by
$$
Z=\frac{e^{-h v / 2 k T}}{1-e^{-h v / k T}}
$$
If the crystal consists of $N_{\mathrm{A}}$ lattice points:
(a) Calculate the Helmholtz function $A$.
(b) Calculate the pressure $P$.
(c) Calculate the entropy $S$.
(d) Express the zero-point energy in terms of $\Theta_{\mathrm{E}}$.

Stanley Enemuo

Numerade Educator

02:24

Problem 1

A substance experiences a phase transition at the temperatures and pressures indicated by the points on the curves in Fig. $14-14$, parts $(a),(b)$, and $(c) .$ Choose any point where the slope is $(d P / d T)_{\lambda}$ in part $(a),(d s / d T)_{\lambda}$ in part $(b)$, and $(d v / d T)_{\lambda}$ in part $(c) .$ Near this point, where $T=T_{\lambda}+\Delta T$, show that:
$$
\frac{c_{V}}{T}=\left(\frac{d s}{d T}\right)_{\lambda}-\frac{\beta}{\kappa}\left(\frac{d v}{d T}\right)_{\lambda}
$$
and
$$
\beta=\frac{1}{\nu}\left(\frac{d v}{d T}\right)_{\lambda}-\kappa\left(\frac{d P}{d T}\right)_{\lambda}
$$

Narayan Hari

Numerade Educator

06:36

Problem 2

In a second-order phase transition, $s^{(i)}=s^{(f)}$ at $(T, P)$, and $s^{(i)}+d s^{(i)}=s^{(f)}+d s^{(f)}$ at $(T+d T, P+d P)$
(a) Prove that
$$
\frac{d P}{d T}=\frac{1}{T v} \frac{c_{P}^{(f)}-c_{P}^{(0)}}{\beta f\rangle-\beta^{(1)}}
$$
In a second-order phase transition, $v^{(i)}=v^{(f)}$ at $(T, P)$, and $v^{(0)}+d u^{(0)}=$ $v^{(f)}+d u^{(f)}$ at $(T+d T, P+d P)$
(b) Prove that
$$
\frac{d P}{d T}=\frac{\beta^{(f)}-\beta^{(i)}}{\kappa^{(f)}-\beta^{(i)}}
$$

Amany Waheeb

Numerade Educator

01:45

Problem 3

For a two-phase system in equilibrium, $P$ is a function of $T$ only; therefore,
$$
\left(\frac{\partial P}{\partial T}\right)_{V}=\left(\frac{\partial P}{\partial T}\right)_{S}=\frac{d P}{d T}
$$
For such a system show that
$$
\frac{C_{V}}{\kappa_{S}}=T V\left(\frac{d P}{d T}\right)^{2}
$$
regardless of the type of transition between the phases.

Manik Pulyani

Numerade Educator

04:56

Problem 4

Using the van der Waals equation of state expressed in "reduced variables," namely,
$$
\left(P_{R}+\frac{3}{v_{R}^{2}}\right)\left(\nu_{R}-\frac{1}{3}\right)=\frac{8}{3} T_{R}
$$
where $P_{R}=P / P_{C}, \nu_{R}=v / v_{C}$, and $T_{R}=T / T_{C}$, calculate values for the following critical-point exponents: $(a) \delta ;(b) \gamma .$ From the molar internal energy for the van der Waals gas, given by
$$
u=c T-\frac{a}{V}
$$
(c) calculate the critical-point exponent $\alpha$.

Farhana Sharmin

Numerade Educator