MikaStars39/nano-eval huggingface dataset:
There is a horizontal dipole antenna installed above the ground with a working wavelength of $\lambda = 40$ m. If the maximum radiation elevation angle of 30° is to be achieved in the plane perpendicular to the antenna, how high should the antenna be erected?

Options:
A. 55m
B. 45m
C. 30m
D. 15m
E. 20m
F. 35m
G. 10m
H. 25m
I. 50m
J. 40m
Answer: E
The aperture diameter of a known parabolic reflector antenna is $D = 3$ m, and the focal length to diameter ratio is $f / D = 0.6$.
If the radiation pattern function of the feed source is $|f(\psi)| = \cos^{2} \psi$, find the aperture efficiency, aperture intercept factor, and the gain factor $g$.

Options:
A. 0.261,0.937,0.245
B. 0.248,0.921,0.240
C. 0.263,0.934,0.241
D. 0.255,0.940,0.243
E. 0.260,0.930,0.248
F. 0.275,0.942,0.259
G. 0.266,0.928,0.250
H. 0.289,0.918,0.267
I. 0.282,0.915,0.245
J. 0.273,0.945,0.252
Answer: A
Assume there is an electric field intensity $E_{0}$ perpendicular to the Earth's surface. A conductor hemisphere with a density $\rho$ and a radius $R$ is placed flat on the ground. Given that the gravitational acceleration on the ground is $g$, and assuming the ground is an infinite conductive plane, what is the minimum value of $E_{0}$ required to lift the hemisphere off the ground?

Options:
A. $\\E_0 > \sqrt{\frac{4 \rho g R}{27 \epsilon_0}}$
B. $\\E_0 > \sqrt{\frac{6 \rho g R}{27 \epsilon_0}}$
C. $\\E_0 > \sqrt{\frac{2 \rho g R}{27 \epsilon_0}}$
D. $\\E_0 > \sqrt{\frac{8 \rho g R}{81 \epsilon_0}}$
E. $\\E_0 > \sqrt{\frac{8 \rho g R}{9 \epsilon_0}}$
F. $\\E_0 > \sqrt{\frac{8 \rho g R}{27 \epsilon_0}}$
G. $\\E_0 > \sqrt{\frac{8 \rho g R}{54 \epsilon_0}}$
H. $\\E_0 > \sqrt{\frac{10 \rho g R}{27 \epsilon_0}}$
I. $\\E_0 > \sqrt{\frac{8 \rho g R}{36 \epsilon_0}}$
J. $\\E_0 > \sqrt{\frac{16 \rho g R}{27 \epsilon_0}}$
Answer: F
A plane electromagnetic wave is incident from a vacuum at an angle of $\theta=4 5^{\circ}$onto a medium with a relative permittivity of $\varepsilon_{r}=2$, with the electric field intensity perpendicular to the plane of incidence. Calculate the reflection coefficient and the transmission coefficient.

Options:
A. $R = \left( \frac{\frac{\sqrt{2}}{3} - \frac{\sqrt{5} \sqrt{3}}{3}}{\frac{\sqrt{2}}{3} + \frac{\sqrt{5} \sqrt{3}}{3}} \right)^2 = \frac{2 - \sqrt{5}}{2 + \sqrt{5}}$
B. $R = \left( \frac{\frac{\sqrt{1}}{2} - \frac{\sqrt{2} \sqrt{4}}{2}}{\frac{\sqrt{1}}{2} + \frac{\sqrt{2} \sqrt{4}}{2}} \right)^2 = \frac{1 - \sqrt{2}}{1 + \sqrt{2}}$
C. $R = \left( \frac{\frac{\sqrt{5}}{3} - \frac{\sqrt{3} \sqrt{1}}{3}}{\frac{\sqrt{5}}{3} + \frac{\sqrt{3} \sqrt{1}}{3}} \right)^2 = \frac{5 - \sqrt{1}}{5 + \sqrt{1}}$
D. $R = \left( \frac{\frac{\sqrt{2}}{2} - \frac{\sqrt{6} \sqrt{3}}{2}}{\frac{\sqrt{2}}{2} + \frac{\sqrt{6} \sqrt{3}}{2}} \right)^2 = \frac{2 - \sqrt{9}}{2 + \sqrt{9}}$
E. $R = \left( \frac{\frac{\sqrt{2}}{4} - \frac{\sqrt{3} \sqrt{6}}{4}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{6}}{4}} \right)^2 = \frac{2 - \sqrt{6}}{2 + \sqrt{6}}$
F. $R = \left( \frac{\frac{\sqrt{4}}{3} - \frac{\sqrt{2} \sqrt{1}}{3}}{\frac{\sqrt{4}}{3} + \frac{\sqrt{2} \sqrt{1}}{3}} \right)^2 = \frac{4 - \sqrt{1}}{4 + \sqrt{1}}$
G. $R = \left( \frac{\frac{\sqrt{3}}{2} - \frac{\sqrt{2} \sqrt{2}}{2}}{\frac{\sqrt{3}}{2} + \frac{\sqrt{2} \sqrt{2}}{2}} \right)^2 = \frac{3 - 2}{3 + 2}$
H. $R = \left( \frac{\frac{\sqrt{3}}{3} - \frac{\sqrt{2} \sqrt{4}}{3}}{\frac{\sqrt{3}}{3} + \frac{\sqrt{2} \sqrt{4}}{3}} \right)^2 = \frac{3 - \sqrt{4}}{3 + \sqrt{4}}$
I. $R = \left( \frac{\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sqrt{3}}{2}}{\frac{\sqrt{2}}{2} + \frac{\sqrt{2} \sqrt{3}}{2}} \right)^2 = \frac{2 - \sqrt{3}}{2 + \sqrt{3}}$
J. $R = \left( \frac{\frac{\sqrt{6}}{5} - \frac{\sqrt{2} \sqrt{5}}{5}}{\frac{\sqrt{6}}{5} + \frac{\sqrt{2} \sqrt{5}}{5}} \right)^2 = \frac{6 - \sqrt{5}}{6 + \sqrt{5}}$
Answer: I
In the paper "On the Electrodynamics of Moving Bodies," where Einstein established the theory of special relativity, he stated that clocks at the Earth's equator run slower than those at the Earth's poles. Assuming the Earth is 5 billion years old and identical clocks were placed at the equator and the poles at the Earth's formation, how much slower is the clock at the equator compared to the clock at the poles now?

Options:
A. 3.7days
B. 1.2days
C. 2.2days
D. 0.5days
E. 4.1days
F. 0.9days
G. 1.8days
H. 5.6days
I. 2.9days
J. 3.4days
Answer: C
Assume $( X_{1}, \ X_{2}, \ \cdots, \ X_{9} )$ and $( Y_{1}, \ Y_{2}, \ \cdots, \ Y_{16} )$ are two independent simple random samples from a population $N ( \mu, 2^{2} )$. Let
$$
Q_{1}=\sum_{i=1}^{9} ( X_{i}-\overline{X} )^{2}, \quad Q_{2}=\sum_{i=1}^{16} ( Y_{i}-\overline{Y} )^{2}
$$
Find the constants $\epsilon$, $b$, and $c$ that satisfy the following equations:
$$
P \left\{\frac{| \overline{Y}-\mu|}{\sqrt{Q_{1}}} \leq a \right\}=0.95, \quad P \left\{b \leq \frac{Q_{1}}{Q_{2}} \leq c \right\}=0.95
$$
What should $a$, $b$, and $c$ be?

Options:
A. $$
a=0. 2 2 0 5
$$
$$
b=0. 1 1 0 7, \; \; c=1. 9 5 8 6
$$
B. $$
a=0. 3 0 0 4
$$
$$
b=0. 1 5 0 2, \; \; c=1. 6 5 8 3
$$
C. $$
a=0. 2 9 0 8
$$
$$
b=0. 1 2 4 5, \; \; c=1. 7 2 8 1
$$
D. $$
a=0. 2 1 5 0
$$
$$
b=0. 1 3 3 3, \; \; c=1. 8 0 0 9
$$
E. $$
a=0. 2 0 3 9
$$
$$
b=0. 1 3 0 1, \; \; c=1. 7 0 6 7
$$
F. $$
a=0. 2 5 0 0
$$
$$
b=0. 1 4 5 0, \; \; c=1. 8 5 0 0
$$
G. $$
a=0. 2 4 0 9
$$
$$
b=0. 1 2 4 9, \; \; c=1. 8 2 5 3
$$
H. $$
a=0. 3 0 9 1
$$
$$
b=0. 1 5 7 8, \; \; c=1. 7 7 4 2
$$
I. $$
a=0. 1 8 9 7
$$
$$
b=0. 1 1 4 3, \; \; c=1. 9 2 3 4
$$
J. $$
a=0. 3 5 0 0
$$
$$
b=0. 1 9 4 2, \; \; c=1. 6 1 0 7
$$
Answer: E
In bag A, there are $n$ white balls and $m$ red balls; in bag B, there are $N$ white balls and $M$ red balls. Now, if one ball is randomly taken from bag A and placed into bag B, and then one ball is randomly drawn from bag B, what is the probability of drawing a white ball?

Options:
A. $$
\frac{m N+n+m M} {( n+m ) ( N+M+1 )}
$$
B. $$
\frac{n N+m+n M} {( n+m ) ( N+M+1 )}
$$
C. $$
\frac{n N+n+m N} {( m+n ) ( N+M+1 )}
$$
D. $$
\frac{m N+m+n N} {( n+m ) ( N+M+1 )}
$$
E. $$
\frac{m N+n+n N} {( m+n ) ( N+M+1 )}
$$
F. $$
\frac{n N+m+n M} {( n+m ) ( M+N+1 )}
$$
G. $$
\frac{m N+n+n M} {( m+n ) ( N+M+1 )}
$$
H. $$
\frac{n N+m+m N} {( n+m ) ( M+N+1 )}
$$
I. $$
\frac{n N+m+n N} {( n+m ) ( M+N+1 )}
$$
J. $$
\frac{n N+n+n M} {( m+n ) ( M+N+1 )}
$$
Answer: C
Let the population $\mathbf{X}$ follow an exponential distribution, with its probability density given by
$$
f(x) = \left\{
\begin{array}{ll}
\frac{1}{\theta} e^{-x / \theta}, & x > 0, \\
0, & x \leq 0, \\
\end{array}
\right.
$$
where $\theta > 0$ is unknown. A sample of size $n$, $X_{1}, X_{2}, \cdots, X_{n}$, is taken from the population. What is the one-sided lower confidence limit for $\theta$ at a confidence level of $1-\alpha$?

Options:
A. $$
\underline{{{{\theta}}}}=\frac{n \overline{{{{X}}}}} {2 \chi_{n}^{2} ( \alpha )}
$$
B. $$
\underline{{{{\theta}}}}=\frac{2 \overline{{{{X}}}}}{n \chi_{n}^{2} ( 2 \alpha )}
$$
C. $$
\underline{{{{\theta}}}}=\frac{\overline{{{{X}}}}} {\chi_{n}^{2} ( n )}
$$
D. $$
\underline{{{{\theta}}}}=\frac{n \overline{{{{X}}}}} {\chi_{n-1}^{2} ( \alpha )}
$$
E. $$
\underline{{{{\theta}}}}=\frac{2 n \overline{{{{X}}}}} {\chi_{n-1}^{2} ( 2 \alpha )}
$$
F. $$
\underline{{{{\theta}}}}=\frac{2 \overline{{{{X}}}}} {\chi_{n-1}^{2} ( n )}
$$
G. $$
\underline{{{{\theta}}}}=\frac{\overline{{{{X}}}}} {\chi_{n-1}^{2} ( 2 \alpha )}
$$
H. $$
\underline{{{{\theta}}}}=\frac{2 n \overline{{{{X}}}}} {\chi_{n}^{2} ( 2 n )}
$$
I. $$
\underline{{{{\theta}}}}=\frac{2 \overline{{{{X}}}}} {\chi_{n}^{2} ( \alpha )}
$$
J. $$
\underline{{{{\theta}}}}=\frac{2 n \overline{{{{X}}}}} {\chi_{n}^{2} ( n )}
$$
Answer: H
The number of doctors in a certain region has increased year by year, with the year-on-year growth rates for the years 1993, 1994, and 1993 being 89%, 18%, and 15%, respectively. By what percentage has the number of doctors in this region increased over the three years in total?

Options:
A. 61.34%
B. 49.18%
C. 58.91%
D. 45.23%
E. 50.32%
F. 4 6. 5 6 %
G. 34.56%
H. 47.29%
I. 39.67%
J. 52.78%
Answer: F
The known van der Waals gas constants for $\mathrm{CO}_{2}$ are $a = 0.3638 \ \mathrm{Pa} \cdot \mathrm{m}^{6} \cdot \mathrm{mol}^{-2}$ and $b = 4.3 \times 10^{-5} \ \mathrm{m}^{3} \cdot \mathrm{mol}^{-1}$, with a boiling point of 194.7 K. When this gas expands through a throttle from $50 \times p^{\Theta}$ to $1 \times p^{\Theta}$, its temperature decreases from the original 298 K to 234 K. To further expand the above $\mathrm{CO}_{2}$ through throttling and decrease its temperature to the boiling point (with the final pressure still at $1 \times p^{\Theta}$), what should the initial pressure be?

Options:
A. $$
2. \, 3 1 1 \, \times\, 1 0^{6} \, \, \mathrm{P a}
$$
B. $$
1. \, 0 8 5 \, \times\, 1 0^{7} \, \, \mathrm{P a}
$$
C. $$
4. \, 9 2 1 \, \times\, 1 0^{7} \, \, \mathrm{P a}
$$
D. $$
6. \, 7 4 3 \, \times\, 1 0^{8} \, \, \mathrm{P a}
$$
E. $$
3. \, 7 6 4 \, \times\, 1 0^{6} \, \, \mathrm{P a}
$$
F. $$
8. \, 1 1 5 \, \times\, 1 0^{6} \, \, \mathrm{P a}
$$
G. $$
5. \, 3 0 2 \, \times\, 1 0^{8} \, \, \mathrm{P a}
$$
H. $$
9. \, 2 4 6 \, \times\, 1 0^{5} \, \, \mathrm{P a}
$$
I. $$
7. \, 5 3 9 \, \times\, 1 0^{4} \, \, \mathrm{P a}
$$
J. $$
2. \, 5 8 9 \, \times\, 1 0^{5} \, \, \mathrm{P a}
$$
Answer: F
At $1000 \, \text{K}, 101325 \, \text{Pa}$, there is a mixture solution of metals A and B, where the amount of substance for metal A is $n_{\mathrm{A}} = 5000 \, \text{mol}$, and for metal B is $n_{\mathrm{B}} = 40 \, \text{mol}$. It is known that the Gibbs free energy of the solution relates to temperature and the amount of substance as follows:
$$
G = n_{\mathrm{A}} G_{\mathrm{m}, \mathrm{A}} + n_{\mathrm{B}} G_{\mathrm{m}, \mathrm{B}} - \left[ \, 0.05774 \, \left( \frac{n_{\mathrm{A}}}{\text{mol}} \right)^{2} + 7.950 \, \left( \frac{n_{\mathrm{B}}}{\text{mol}} \right)^{3} + 2.385 \, \left( \frac{T}{\text{K}} \right) \, \right] \, \mathrm{\&}
$$
If this solution is mixed with slag, assuming the slag can be regarded as an ideal solution, and the mole fraction of B in the slag is $x_{\mathrm{B}} = 0.001$. Calculate the activity of B in the metallic liquid.

Options:
A. $$
0. 0 1 0 \ 0 5
$$
B. $$
0. 0 1 1 \ 0 5
$$
C. $$
0. 0 1 0 \ 1 5
$$
D. $$
0. 0 1 0 \ 2 5
$$
E. $$
0. 0 1 0 \ 1 6
$$
F. $$
0. 0 0 9 \ 1 5
$$
G. $$
0. 0 1 0 \ 1 4
$$
H. $$
0. 0 1 2 \ 1 5
$$
I. $$
0. 0 2 0 \ 1 5
$$
J. $$
0. 0 3 0 \ 1 5
$$
Answer: C
Suppose a newly synthesized organic compound $\mathrm{R}$ has mass fractions of carbon, hydrogen, and oxygen of $w_{{c}} = 0.632$, $w_{\mathrm{H}} = 0.088$, and $w_{0} = 0.280$ respectively. Now, 0.0702 g of this organic compound is dissolved in $0.804 g$ of camphor, and its freezing point is lowered by $15.3 K$ compared to pure camphor. Calculate the molar mass of the organic compound and its chemical formula. It is known that the freezing point depression constant of camphor is $K_{\mathrm{f}} = 40.0 ~ \text{K} \cdot \text{mol}^{-1} \cdot \text{kg}^{-1}$ (Due to the large freezing point depression constant of camphor, even though the amount of solute used is small, the freezing point depression value is still large, and compared to boiling point elevation experiments, the accuracy is higher).

Options:
A. $$
n=4,\mathrm{G}_{1 5} \mathrm{H}_{1 8} \mathrm{O}_{5}
$$
B. $$
n=3,\mathrm{G}_{1 4} \mathrm{H}_{1 8} \mathrm{O}_{2}
$$
C. $$
n=3,\mathrm{G}_{1 6} \mathrm{H}_{1 5} \mathrm{O}_{4}
$$
D. $$
n=4,\mathrm{G}_{1 2} \mathrm{H}_{2 0} \mathrm{O}_{4}
$$
E. $$
n=5,\mathrm{G}_{1 0} \mathrm{H}_{2 4} \mathrm{O}_{2}
$$
F. $$
n=4,\mathrm{G}_{1 1} \mathrm{H}_{2 1} \mathrm{O}_{3}
$$
G. $$
n=2,\mathrm{G}_{1 7} \mathrm{H}_{1 4} \mathrm{O}_{6}
$$
H. $$
n=5,\mathrm{G}_{1 0} \mathrm{H}_{2 2} \mathrm{O}_{3}
$$
I. $$
n=5,\mathrm{G}_{1 1} \mathrm{H}_{2 0} \mathrm{O}_{4}
$$
J. $$
n=6,\mathrm{G}_{1 3} \mathrm{H}_{1 6} \mathrm{O}_{5}
$$
Answer: D
Considering a particle system composed of an extremely large number of three-dimensional translational particles moving within a cubic container with side length $a$, the relationship between the system's volume, particle mass, and temperature is given by $\frac{h^{2}}{8 m a^{2}} = 0.10 k T_{\circ}$. The energies of the two available energy levels are $\varepsilon_{1} = \frac{9 h^{2}}{4 m a^{2}}$ and $\varepsilon_{2} = \frac{27 h^{2}}{8 m a^{2}}$. Calculate the ratio of the number of particles at these two energy levels ($N_{1} / N_{2}$).

Options:
A. 0.78
B. 2.50
C. 3.67
D. 1.45
E. 2.13
F. 1.65
G. 0.92
H. 1.84
I. 3.25
J. 2.71
Answer: H
The gas-phase reaction $\mathrm{A+B=2C+D}$ occurs at 298 K in a constant-volume container. The initial partial pressures of $\mathrm{A}$ and $\mathrm{B}$ are both 101.325 kPa. When the reaction reaches equilibrium, the equilibrium partial pressures of A and B are both ${\frac{1}{3}} \times 101.325 \, \mathrm{kPa}$. Initially, the container does not contain C and D. Thus, the equilibrium constant $K_{\varepsilon} / \, (\mathrm{mol} \cdot \, \mathrm{dm}^{-3})$ of the reaction at 298 K is _____.

Options:
A. $$
4. 3 1 \times1 0^{-3}
$$
B. $$
2. 7 2 \times 1 0^{-3}
$$
C. $$
7. 9 6 \times 1 0^{-3}
$$
D. $$
8. 4 7 \times 1 0^{-3}
$$
E. $$
1. 5 4 \times 1 0^{-3}
$$
F. $$
6. 7 1 \times 1 0^{-3}
$$
G. $$
5. 9 8 \times 1 0^{-3}
$$
H. $$
3. 6 4 \times 1 0^{-3}
$$
I. $$
3. 1 2 \times 1 0^{-3}
$$
J. $$
2. 4 5 \times 1 0^{-3}
$$
Answer: A
The equilibrium constant of a certain real gas reaction is expressed in terms of fugacity as $K_{f}^{\Theta}$, so $K_{f}^{\Theta}$ is related to which of the following physical quantities: ____.

Options:
A. Volume
B. Viscosity
C. Specific Heat
D. Enthalpy
E. Concentration
F. Density
G. Temperature
H. Pressure
I. Entropy
J. Surface Area
Answer: G
Divalent ionic crystal. BaO has a NaCl structure. Try to estimate the binding energy per molecule of the hypothetical ions $\mathbf{Ba}^{+} \mathbf{O}^{-}$ and $\mathbf{Ba}^{++} \mathbf{O}^{--}$ compared to when they are separated into neutral atoms. The observed nearest neighbor distance is $R_{0} = 2.76 \mathbf{A}$; the first and second ionization energies of Ba are $5.19 \text{eV}$ and $9.96 \text{eV}$; the first and second electron affinities added to a neutral oxygen atom are $1.5 \text{eV}$ and $-9.0 \text{eV}$. The first electron affinity of a neutral oxygen atom is the energy released in the reaction $O + e \to O^{-}$. The second electron affinity is the energy released in the reaction $O^{-} + e \to O^{--}$. Assuming both forms of barium oxide have the same $R$, and neglecting repulsive energy, which valence state is predicted to appear?

Options:
A. $\mathbf{B} \mathbf{a}^{+} \mathbf{O}^{-}$ is a more likely combination state.
B. $\mathbf{B} \mathbf{a}^{4+} \mathbf{O}^{4-}$ is a more stable oxidation condition.
C. $\mathbf{B} \mathbf{a}^{++} \mathbf{0}^{--}$ is a more favorable oxidation state.
D. $\mathbf{B} \mathbf{a}^{0} \mathbf{O}^{-1}$ is a more stable ionic state.
E. $\mathbf{B} \mathbf{a}^{+2} \mathbf{0}^{0}$ is a stronger bound state.
F. $\mathbf{B} \mathbf{a}^{1+} \mathbf{O}^{0}$ forms a more feasible duplet state.
G. $\mathbf{B} \mathbf{a}^{2+} \mathbf{O}^{0}$ reflects a preferable ionization state.
H. $\mathbf{B} \mathbf{a}^{} \mathbf{O}^{}$ is an equally potent oxidation state.
I. $\mathbf{B} \mathbf{a}^{3+} \mathbf{O}^{3-}$ is an energetically preferred state.
J. $\mathbf{B} \mathbf{a}^{0} \mathbf{O}^{0}$ is a less practical ionized state.
Answer: C
The spin of the helium-3 atom $\mathrm{He}^{3}$ is $1/2$ and it is a fermion. The density of liquid $\mathrm{He}^{3}$ near absolute zero is $0.081 \mathrm{g/cm}^{3}$. The Fermi energy $e_{F}$ is () and the Fermi temperature $T_{F}$ is ().

Options:
A. $$
\approx4. 0 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx6. \mathrm{0 K}
$$
B. $$
\approx6. 1 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx5. \mathrm{2 K}
$$
C. $$
\approx6. 5 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx5. \mathrm{7 K}
$$
D. $$
\approx3. 6 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{3 K}
$$
E. $$
\approx4. 8 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx2. \mathrm{9 K}
$$
F. $$
\approx5. 7 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx3. \mathrm{4 K}
$$
G. $$
\approx7. 5 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx3. \mathrm{8 K}
$$
H. $$
\approx4. 3 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{9 K}
$$
I. $$
\approx3. 9 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx6. \mathrm{1 K}
$$
J. $$
\approx8. 2 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{5 K}
$$
Answer: H
A two-dimensional metal contains a monovalent atom in a simple rectangular unit cell ( a=2A, b=4A \). The radius of the free electron Fermi circle (in cm) is ().

Options:
A. $$
0. 5 6 7 \times1 0^{8} \mathrm{c m}^{-1}
$$
B. $$
0. 9 3 2 \times1 0^{8} \mathrm{c m}^{-1}
$$
C. $$
0. 8 4 9 \times1 0^{8} \mathrm{c m}^{-1}
$$
D. $$
0. 8 0 1 \times1 0^{8} \mathrm{c m}^{-1}
$$
E. $$
0. 8 8 6 \times1 0^{8} \mathrm{c m}^{-1}
$$
F. $$
0. 1 1 2 \times1 0^{8} \mathrm{c m}^{-1}
$$
G. $$
1. 0 0 0 \times1 0^{8} \mathrm{c m}^{-1}
$$
H. $$
0. 6 5 5 \times1 0^{8} \mathrm{c m}^{-1}
$$
I. $$
0. 9 9 1 \times1 0^{8} \mathrm{c m}^{-1}
$$
J. $$
0. 7 2 4 \times1 0^{8} \mathrm{c m}^{-1}
$$
Answer: E
Recent optical research has discovered that organic conductors have a plasma frequency of $\omega_{p}=1.30 \times 10^{15} \mathrm{S}^{-1}$, while the electron relaxation time at room temperature is $t=2.83 \times 10^{-16} \mathrm{S}$. Through analysis of the crystal structure and chemical structure, the concentration of conduction electrons is found to be $4.7 \times 10^{21} \mathrm{cm}^{-3}$. Calculate the effective mass of the electron $m$.

Options:
A. $$ m \approx7. 2 $$
B. $$ m \approx3. 6 $$
C. $$ m \approx6. 8 $$
D. $$ m \approx9. 5 $$
E. $$
m \approx8. 8
$$
F. $$ m \approx5. 6 $$
G. $$ m \approx5. 7 $$
H. $$ m \approx6. 3 $$
I. $$ m \approx4. 9 $$
J. $$ m \approx7. 4 $$
Answer: E
It is known that sodium chloride is a cubic crystal with a relative molecular mass of 58.46 and a density of $2.167 \times 10^{3} \, \mathrm{kg} \cdot \mathrm{m}^{-3}$ at room temperature. Try calculating its lattice constant, which is ().

Options:
A. $$
\mathfrak{a}=5. 6 1 \ \overset{\circ} {\mathrm{A}}
$$
B. $$
\mathfrak{a}=5. 4 5 \ \overset{\circ} {\mathrm{A}}
$$
C. $$
\mathfrak{a}=5. 7 1 \ \overset{\circ} {\mathrm{A}}
$$
D. $$
\mathfrak{a}=5. 5 9 \ \overset{\circ} {\mathrm{A}}
$$
E. $$
\mathfrak{a}=5. 8 0 \ \overset{\circ} {\mathrm{A}}
$$
F. $$
\mathfrak{a}=5. 7 8 \ \overset{\circ} {\mathrm{A}}
$$
G. $$
\mathfrak{a}=5. 4 8 \ \overset{\circ} {\mathrm{A}}
$$
H. $$
\mathfrak{a}=5. 6 9 \ \overset{\circ} {\mathrm{A}}
$$
I. $$
\mathfrak{a}=5. 6 3 \ \overset{\circ} {\mathrm{A}}
$$
J. $$
\mathfrak{a}=5. 5 5 \ \overset{\circ} {\mathrm{A}}
$$
Answer: I
Assume the zero-point vibrational energy of each oscillator in a crystal is $\frac{1}{2} \hbar\omega$. Use the Debye model to find the zero-point vibrational energy of the crystal, which is ().

Options:
A. $$
\frac{5} {6} N \hbar\omega_{\mathrm{m}}
$$
B. $$
\frac{4} {5} N \hbar\omega_{\mathrm{m}}
$$
C. $$
\frac{3} {4} N \hbar\omega_{\mathrm{m}}
$$
D. $$
\frac{9} {8} N \hbar\omega_{\mathrm{m}}
$$
E. $$
\frac{3} {5} N \hbar\omega_{\mathrm{m}}
$$
F. $$
\frac{7} {6} N \hbar\omega_{\mathrm{m}}
$$
G. $$
\frac{1} {2} N \hbar\omega_{\mathrm{m}}
$$
H. $$
\frac{2} {3} N \hbar\omega_{\mathrm{m}}
$$
I. $$
\frac{5} {8} N \hbar\omega_{\mathrm{m}}
$$
J. $$
\frac{7} {8} N \hbar\omega_{\mathrm{m}}
$$
Answer: D
When the inscribed sphere of the first Brillouin zone is a Fermi sphere, the average number of free electrons per atom that corresponds to each in the body-centered cubic structure is approximately (); when the circumscribed sphere of the first Brillouin zone is a Fermi sphere, the average number when of free electrons per atom that corresponds to each in the body-centered cubic structure is approximately ().

Options:
A. $1.75$; $4.32$
B. $1.50$; $4.05$
C. $1.45$; $4.29$
D. $1.59$; $4.31$
E. $1.12$; $3.89$
F. $1.48$; $4.19$
G. $1.36$; $4.23$
H. $1.51$; $4.07$
I. $1.47$; $4.52$
J. $1.62$; $3.98$
Answer: F
Given that the atmospheric pressure gauge reads 755 mmHg and the absolute pressure is 0.5 bar, the corresponding vacuum level (mbar) is determined is().

Options:
A. 72.4kPa.
B. 48.0kPa.
C. 62.1mbar.
D. 69.8kPa.
E. 78.5mbar.
F. 50.6kPa.
G. 45.3kPa.
H. 66.2mbar.
I. 59.4kPa.
J. 55.9kPa.
Answer: F
A boiler for heating purposes has an evaporation rate of 2 tons per hour. The enthalpy of the feedwater entering the boiler is \( h_{1} = 210 \, \text{kJ}/\text{kg} \). The enthalpy of the steam leaving the boiler is \( h_{2} = 2768 \, \text{kJ}/\text{kg} \). Given that the calorific value of the coal is 23,000 kJ/kg and the efficiency of the boiler is 70.96%, calculate the coal consumption per hour

Options:
A. $$
3 1 8 \ \mathrm{k g} / h
$$
B. $$
4 6 0 \ \mathrm{k g} / h
$$
C. $$
4 2 5 \ \mathrm{k g} / h
$$
D. $$
3 5 0 \ \mathrm{k g} / h
$$
E. $$
2 8 0 \ \mathrm{k g} / h
$$
F. $$
3 7 5 \ \mathrm{k g} / h
$$
G. $$
2 5 0 \ \mathrm{k g} / h
$$
H. $$
5 3 6 \ \mathrm{k g} / h
$$
I. $$
4 0 0 \ \mathrm{k g} / h
$$
J. $$
2 1 0 \ \mathrm{k g} / h
$$
Answer: A
For the initial state where the pressure $p_{1}=1 \times 10^{3}$ Pa, temperature $t_{1}=30^{\circ} \ \mathrm{C}$, and mass of air is 1 kg, compress the air to $p_{2}=10 \times 10^{5} \ \mathrm{Pa}$. Find the temperature and the technical work consumed in the reversible polytropic compression process when $n=1.3$

Options:
A. $$
- 3 1 0 \mathrm{~ k J} / \mathrm{k g}
$$
B. $$
- 2 5 5 \mathrm{~ k J} / \mathrm{k g}
$$
C. $$
- 2 5 0 \mathrm{~ k J} / \mathrm{k g}
$$
D. $$
- 2 7 0 \mathrm{~ k J} / \mathrm{k g}
$$
E. $$
- 2 4 0 \mathrm{~ k J} / \mathrm{k g}
$$
F. $$
- 2 8 0 \mathrm{~ k J} / \mathrm{k g}
$$
G. $$
- 2 6 0 \mathrm{~ k J} / \mathrm{k g}
$$
H. $$
- 2 6 5 \mathrm{~ k J} / \mathrm{k g}
$$
I. $$
- 3 0 0 \mathrm{~ k J} / \mathrm{k g}
$$
J. $$
- 2 7 5 \mathrm{~ k J} / \mathrm{k g}
$$
Answer: H
A certain boiler burns 500 kg of coal per hour. It is estimated that each kilogram of coal, when burned, can produce 10 m³ of flue gas under standard conditions. The pressure of the flue gas at the chimney outlet is measured to be $0.1 MPa$, with a temperature of $200°C$, and the gas velocity is \( $c = 3 \, \mathrm{m/s}$ \). Given that the chimney cross-section is circular, what is the inner diameter of the chimney outlet?

Options:
A. $$
1. 7 3 5 m
$$
B. $$
1. 3 1 2 m
$$
C. $$
0. 8 5 3 m
$$
D. $$
0. 9 5 1 m
$$
E. $$
2. 1 6 7 m
$$
F. $$
1. 2 4 6 m
$$
G. $$
2. 5 4 8 m
$$
H. $$
0. 6 4 2 m
$$
I. $$
1. 0 1 7 m
$$
J. $$
1. 5 3 8 m
$$
Answer: I
A mixture of nitrogen and carbon dioxide, at a temperature of $40^{\circ} \ \mathrm{C}$and a pressure of \($5 \times 10^5 \ \mathrm{Pa}$\), has a specific volume of \($0.166 \ \mathrm{m}^3/\mathrm{kg}$\). Determine the mass fraction of the mixture

Options:
A. 0.707,0.293
B. 0.550,0.450
C. 0.650,0.350
D. 0.620,0.380
E. 0.480,0.520
F. 0.490,0.510
G. 0.573,0.427
H. 0.500,0.500
I. 0.680,0.320
J. 0.600,0.400
Answer: A
The total entropy change in the mixing process of water with mass $m$ and temperature $T_{1}$, and water with mass $m$ and temperature $T_{2}$, in an adiabatic container at constant pressure,

Options:
A. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{(T_{1}-T_{2})^{2}} {T_{1} T_{2}}
$$
B. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{T_{1}+T_{2}} {2 \, \sqrt{T_{1} \, T_{2}}}
$$
C. $$
\Delta S=2 m c_{p} \frac{T_{1}-T_{2}} {T_{1}+T_{2}}
$$
D. $$
\Delta S=\frac{m c_{p}} {2} \mathrm{l n} \, \frac{T_{1}+T_{2}} {T_{1} T_{2}}
$$
E. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{\sqrt{T_{1}+T_{2}}} {2 \, T_{1} T_{2}}
$$
F. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{T_{1} T_{2}} {T_{1}+T_{2}}
$$
G. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{T_{1} T_{2}} {2 \, \sqrt{T_{1}+T_{2}}}
$$
H. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{T_{1}+T_{2}} {4 \, T_{1} T_{2}}
$$
I. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{T_{1}+T_{2}} {T_{1} T_{2}}
$$
J. $$
\Delta S=2 m c_{p} \mathrm{l n} \, \frac{(T_{1}+T_{2})^{2}} {4 \, T_{1} T_{2}}
$$
Answer: B
Given that the air's temperature is $20^{\circ} \ \mathrm{C}$ and the relative humidity is $\varphi_{1}=0.60$, with an atmospheric pressure of 0.1 MPa. The air is heated in a heater to $t_{2}=50^{\circ} \ \mathrm{C}$ before entering the drying chamber. When discharged from the drying chamber, the temperature is $t_{3}=30^{\circ} \ \mathrm{C}$. Determine the relative humidity $\varphi_{3}$ of the moist air when leaving the drying chamber, and the amount of air and heating required to evaporate 1 $\mathbf{kg}$ of moisture

Options:
A. $$
0. 6 6
$$
$$
1 3 4 \ \mathrm{k g}
$$
$$
3 2 9 7 \ \mathrm{k J}
$$
B. $$
0. 8 0
$$
$$
1 8 4 \ \mathrm{k g}
$$
$$
3 5 4 6 \ \mathrm{k J}
$$
C. $$
0. 6 9
$$
$$
1 6 2 \ \mathrm{k g}
$$
$$
3 8 9 0 \ \mathrm{k J}
$$
D. $$
0. 4 3
$$
$$
1 4 5 \ \mathrm{k g}
$$
$$
3 8 0 1 \ \mathrm{k J}
$$
E. $$
0. 4 8
$$
$$
1 4 0 \ \mathrm{k g}
$$
$$
3 9 8 2 \ \mathrm{k J}
$$
F. $$
0. 7 5
$$
$$
1 1 9 \ \mathrm{k g}
$$
$$
4 1 0 8 \ \mathrm{k J}
$$
G. $$
0. 6 3
$$
$$
1 2 3 \ \mathrm{k g}
$$
$$
3 7 7 6 \ \mathrm{k J}
$$
H. $$
0. 5 5
$$
$$
1 3 9 \ \mathrm{k g}
$$
$$
4 0 2 3 \ \mathrm{k J}
$$
I. $$
0. 5 0
$$
$$
1 2 8 \ \mathrm{k g}
$$
$$
3 3 3 5 \ \mathrm{k J}
$$
J. $$
0. 7 1
$$
$$
1 0 5 \ \mathrm{k g}
$$
$$
3 4 6 9 \ \mathrm{k J}
$$
Answer: G
The air temperature before compression of a certain diesel engine is $90^{\circ}C$; the ignition temperature of diesel is $400^{\circ}C$. The question is: what is the minimum compression ratio needed to ignite the diesel? Given that the heat capacity ratio of air $k=1.40$.

Options:
A. 8.47
B. 6.31
C. 3.20
D. 5.89
E. 2.76
F. 9.63
G. 4.68
H. 7.24
I. 3.95
J. 5.12
Answer: G
The thermometer sheath used to measure the air temperature inside the gas storage tank is made of steel, with a thermal conductivity of $\lambda = 45 \ \ \mathrm{W / (m \cdot K)}$. The sheath wall thickness is $\delta = 1.5 \ \mathrm{mm}$, and its length is $H = 120 \ \mathrm{mm}$. The temperature at the end of the thermometer indicating sheath is 80°C, while the temperature at the other end, where the sheath connects to the gas storage tank, is 40°C. The known surface heat transfer coefficient for convective heat exchange between the sheath and the air inside the tank is $5\mathrm{W / (m^{2} \cdot K)}$. Try to determine the temperature measurement error caused by the heat conduction through the sheath

Options:
A. $$
- 3 9 . 2^{\circ} C
$$
B. $$
5 5 . 4^{\circ} C
$$
C. $$
4 4 . 6^{\circ} C
$$
D. $$
- 2 5 . 3^{\circ} C
$$
E. $$
- 6 8. 7^{\circ} C
$$
F. $$
6 2 . 9^{\circ} C
$$
G. $$
7 5 . 1^{\circ} C
$$
H. $$
- 1 4 . 7^{\circ} C
$$
I. $$
3 1 . 4^{\circ} C
$$
J. $$
- 5 0 . 8^{\circ} C
$$
Answer: E
For a certain blackbody, the wavelength of maximum spectral radiance $\lambda_{\operatorname* {min}} = 5.8 \; \; \mu\mathrm{m}$. Calculate the fraction of radiated energy of this blackbody within the wavelength range of $1 \sim 5 \mu\mathrm{m}$

Options:
A. $$
1 0. 4 5 8 \%
$$
B. $$
2 3. 7 8 3 \%
$$
C. $$
4 9. 1 2 3 \%
$$
D. $$
7 5. 8 4 6 \%
$$
E. $$
8 1. 5 9 0 \%
$$
F. $$
9 7. 3 4 9 \%
$$
G. $$
1 2. 0 3 4 \%
$$
H. $$
1 6. 1 9 5 \%
$$
I. $$
5 8. 2 9 1 \%
$$
J. $$
3 2. 6 7 2 \%
$$
Answer: H
If using double-layer glass windows, the size of the glass window, the thickness of the glass, and the convection heat transfer conditions inside and outside are the same as in questions 1-3. The air layer thickness between the double glass is 5mm, and the air in the layer is completely still. The thermal conductivity of the air is $\lambda=0.025 \ \ \mathrm{W / (m \cdot K)}$. Calculate the heat loss of the glass window and the thermal resistance of the air layer

Options:
A. $$
3 7. 5 0 W
$$
$$
0. 4 K / W
$$
B. $$
4 0. 2 0 W
$$
$$
0. 1 K / W
$$
C. $$
3 5. 8 0 W
$$
$$
0. 7 K / W
$$
D. $$
3 9. 9 5 W
$$
$$
0. 2 K / W
$$
E. $$
4 4. 4 4 W
$$
$$
0. 2 K / W
$$
F. $$
4 3. 3 9 W
$$
$$
0. 6 K / W
$$
G. $$
4 1. 0 0 W
$$
$$
0. 2 K / W
$$
H. $$
3 6. 6 5 W
$$
$$
0. 5 K / W
$$
I. $$
4 2. 1 5 W
$$
$$
0. 1 K / W
$$
J. $$
3 8. 8 8 W
$$
$$
0. 3 K / W
$$
Answer: D
Using the $k_{*} \left( \lambda\!=\! 0.1542 \ \mathrm{nm} \right)$ of $Cu$, the first three spectral lines measured in the X-ray diffraction spectrum of Cr are $2 \theta=44.4^{\circ}$, 64.6°, and 81.8°. If the lattice constant for (bcc) Cr is $a = 0.2885 \, \mathrm{nm}$, what are the Miller indices corresponding to these spectral lines?

Options:
A. $$
\approx1, \approx4, \approx6
$$
B. $$
\approx3, \approx5, \approx7
$$
C. $$
\approx1, \approx5, \approx6
$$
D. $$
\approx2, \approx4, \approx6
$$
E. $$
\approx2, \approx2, \approx5
$$
F. $$
\approx2, \approx4, \approx5
$$
G. $$
\approx3, \approx4, \approx6
$$
H. $$
\approx1, \approx3, \approx5
$$
I. $$
\approx2, \approx3, \approx6
$$
J. $$
\approx0, \approx4, \approx7
$$
Answer: D
Austenitic steel with a w(Mo) of 12.3% and w(C) of 1.34%, a lattice constant of 0.3624 nm, and a density of 7.83 g/cm³. The relative atomic masses of C, Fe, and Mn are 12.01, 55.85, and 54.94, respectively. What is the type of this solid solution?

Options:
A. This solid solution is a C-substitutional, Mn-interstitial solid solution.
B. This solid solution is a Fe-substitutional, Mn-interstitial solid solution.
C. This solid solution is a Mn-interstitial, Fe-substitutional solid solution.
D. This solid solution is a Mn-substitutional, C-interstitial solid solution.
E. This solid solution is a Mo-substitutional, C-interstitial solid solution.
F. This solid solution is a Fe-interstitial, Mo-substitutional solid solution.
G. This solid solution is a Fe-interstitial, C-substitutional solid solution.
H. This solid solution is a Mo-interstitial, Fe-substitutional solid solution.
I. This solid solution is a C-substitutional, Fe-interstitial solid solution.
J. This solid solution is a C-interstitial, Mn-substitutional solid solution.
Answer: J
The known number-average relative molecular mass of linear polytetrafluoroethylene is $5 \times 10^{5}$. The C-C bond length is 0.154 nm, and the bond angle $\theta$ is $109^{\circ}$. What are its total chain length $L$ and root mean square length?

Options:
A. 1389.325(nm),17.3(nm)
B. 1296.471(nm),12.7(nm)
C. 1458.359(nm),16.1(nm)
D. 1089.734(nm),18.5(nm)
E. 1178.562(nm),14.2(nm)
F. 1234.567(nm),11.5(nm)
G. 1421.984(nm),13.9(nm)
H. 1253.738(nm),15.4(nm)
I. 1106.892(nm),19.6(nm)
J. 1347.652(nm),16.8(nm)
Answer: H
In a diffusion experiment with a certain crystal, it was found that at $500^{\circ} \ \mathrm{C}$, 1 out of $10^{10}$ atoms has enough activation energy to jump out of its equilibrium position and into an interstitial position; at $600^{\circ} \ \mathrm{C}$, this ratio increases to $10^{9}⁻⁹$. At $700^{\circ} \ \mathrm{C}$, what proportion of atoms have sufficient energy?

Options:
A. $$
9 \times1 0^{-8}
$$
B. $$
7 \times1 0^{-10}
$$
C. $$
4 \times1 0^{-9}
$$
D. $$
6 \times1 0^{-9}
$$
E. $$
5 \times1 0^{-8}
$$
F. $$
3 \times1 0^{-7}
$$
G. $$
5 \times1 0^{-10}
$$
H. $$
2 \times1 0^{-7}
$$
I. $$
8 \times1 0^{-6}
$$
J. $$
1 \times1 0^{-8}
$$
Answer: D
If an additional (111) plane is inserted causing a 1° tilt small-angle grain boundary within $\alpha$-Fe, what is the average dislocation spacing?

Options:
A. $$
3 6. 5 0 0 2 ( \mathrm{n m} )
$$
B. $$
3 3. 9 9 8 3 ( \mathrm{n m} )
$$
C. $$
2 9. 4 1 5 9 ( \mathrm{n m} )
$$
D. $$
2 6. 4 3 5 1 ( \mathrm{n m} )
$$
E. $$
3 1. 4 2 5 7 ( \mathrm{n m} )
$$
F. $$
1 8. 9 6 1 5 ( \mathrm{n m} )
$$
G. $$
2 1. 7 7 6 4 ( \mathrm{n m} )
$$
H. $$
2 5. 7 3 2 8 ( \mathrm{n m} )
$$
I. $$
3 2. 0 8 4 5 ( \mathrm{n m} )
$$
J. $$
2 4. 9 8 0 6 ( \mathrm{n m} )
$$
Answer: F
For a $bcc$ crystal, the critical resolved shear stress for the $(1 \bar{1} 0) [1 1 1]$ slip system is $60MPa$. Determine how much stress must be applied in the [001] and [010] directions to induce slip?

Options:
A. $$
45.3,
1 4 6. 9 7 ( \mathrm{M P a} )
$$
B. $$
\infty,
1 4 6. 9 7 ( \mathrm{M P a} )
$$
C. $$
\infty,
1 7 8. 2 4 ( \mathrm{M P a} )
$$
D. $$
\infty,
2 5 6. 1 8 ( \mathrm{M P a} )
$$
E. $$
77.8,
1 4 6. 9 7 ( \mathrm{M P a} )
$$
F. $$
\infty,
1 9 0. 4 5 ( \mathrm{M P a} )
$$
G. $$
\infty,
1 0 2. 3 4 ( \mathrm{M P a} )
$$
H. $$
\infty,
2 7 3. 6 8 ( \mathrm{M P a} )
$$
I. $$
\infty,
3 0 9. 5 1 ( \mathrm{M P a} )
$$
J. $$
32.5,
1 4 6. 9 7 ( \mathrm{M P a} )
$$
Answer: B
After spheroidizing annealing, all the cementite in 40 steel is in the form of spheres with a radius of 10 μm, uniformly distributed on the α-Fe matrix. Given that the shear modulus $G = 7.9 \times 10^{4} \mathrm{~MPa}$ for Fe, and the lattice constant of α-Fe is $a = 0.28 \mathrm{~nm}$, what is the shear strength of 40 steel?

Options:
A. 0.590(MPa)
B. 0.465(MPa)
C. 0.475(MPa)
D. 0.530(MPa)
E. 0.490(MPa)
F. 0.430(MPa)
G. 0.550(MPa)
H. 0.620(MPa)
I. 0.510(MPa)
J. 0.380(MPa)
Answer: B
Calculate the change in the melting point of tin when the pressure increases to \($500 \times 10^{5} \, \mathrm{Pa}$\) is_____. It is known that under \($10^{5} \, \mathrm{Pa}$\), the melting point of tin is 505 K, the heat of fusion is $7196 J/mol$, the molar mass is \($118.8 \times 10^{-3}$ \, \mathrm{kg/mol}\), the density of solid tin is \($7.30 \times 10^{3} \, \mathrm{kg/m}^{3}$\), and the volume change upon melting is +2.7%.

Options:
A. $1.54(K)$
B. $1.47(K)$
C. $1.78(K)$
D. $1.62(K)$
E. $1.59(K)$
F. $1.51(K)$
G. $1.42(K)$
H. $1.73(K)$
I. $1.49(K)$
J. $1.66(K)$
Answer: A
On the crystal slip plane, there is a dislocation loop. Given a shear stress $\tau$ applied in the direction of its Burgers vector, what is the minimum radius required for the dislocation loop to be stable within the crystal?

Options:
A. $$
r_{c}={\frac{a^2 \tau} {G b}}
$$
B. $$
r_{c}={\frac{G b} {a \tau}}
$$
C. $$
r_{c}={\frac{a b} {G \tau}}
$$
D. $$
r_{c}={\frac{b a \tau} {G}}
$$
E. $$
r_{c}={\frac{a \tau} {b G}}
$$
F. $$
r_{c}={\frac{a G b} {\tau}}
$$
G. $$
r_{c}={\frac{a G^2} {b \tau}}
$$
H. $$
r_{c}={\frac{G^2 b} {a \tau}}
$$
I. $$
r_{c}={\frac{b \tau} {a G}}
$$
J. $$
r_{c}={\frac{G b \tau} {a}}
$$
Answer: F
Diamond is a crystalline structure of carbon with a lattice constant of $a = 0.357$ nm. When it transforms into graphite (with a density of $2.25 \text{ g/cm}^3$), what is the change in its volume?

Options:
A. Compression 8 2.6%
B. Expansion 3 7.5%
C. Expansion 4 6.3%
D. Inflation 5 5.8%
E. Shrinkage 4 6.7%
F. Contraction 9 1.4%
G. Recession 10 3.9%
H. Inflation 6 4.1%
I. Deflation 8 4.9%
J. Reduction 7 3.2%
Answer: D
It is known that pure titanium has two allotropic forms: the low-temperature stable hexagonal close-packed structure $\alpha-T_{1}$ and the high-temperature stable body-centered cubic structure $\beta-\mathrm{Ti}$. The allotropic transformation temperature is $882.5^\circ C$. What are the interplanar spacings of the crystal planes (112) and (001) in pure titanium at room temperature $(20^\circ C)$ and $900^\circ C$? (Given: $a_{\alpha}^{\mathtt{20^\circ C}}=0.2951~\mathrm{nm}$, $c_{\alpha}^{\mathtt{20^\circ C}}=0.4679~\mathrm{nm}$, and $a_{\beta}^{\mathtt{20^\circ C}}=0.3307~\mathrm{nm}$).

Options:
A. $$0.133~\mathrm{nm},~0.1664~\mathrm{nm}$$
B. $$0.140~\mathrm{nm},~0.1615~\mathrm{nm}$$
C. $$0.130~\mathrm{nm},~0.1640~\mathrm{nm}$$
D. $$0.137~\mathrm{nm},~0.1622~\mathrm{nm}$$
E. $$0.145~\mathrm{nm},~0.1723~\mathrm{nm}$$
F. $$0.128~\mathrm{nm},~0.1587~\mathrm{nm}$$
G. $$0.135~\mathrm{nm},~0.1653~\mathrm{nm}$$
H. $$0.138~\mathrm{nm},~0.1691~\mathrm{nm}$$
I. $$0.142~\mathrm{nm},~0.1637~\mathrm{nm}$$
J. $$0.136~\mathrm{nm},~0.1680~\mathrm{nm}$$
Answer: G
A light source emits monochromatic visible light with a wavelength of \(6 \times 10^{-7} \, \text{m}\) at a power of 0.6W. The approximate number of photons emitted per second is ().

Options:
A. $$
3 \times1 0^{1 9}
$$
B. $$
9 \times1 0^{1 6}
$$
C. $$
6 \times1 0^{1 9}
$$
D. $$
8 \times1 0^{1 7}
$$
E. $$
5 \times1 0^{1 6}
$$
F. $$
7 \times1 0^{1 8}
$$
G. $$
2 \times1 0^{1 8}
$$
H. $$
4 \times1 0^{1 7}
$$
I. $$
1 \times1 0^{1 8}
$$
J. $$
5 \times1 0^{1 8}
$$
Answer: G
According to the Bohr model of the hydrogen atom, the radius of an excited state with $n=100$ is approximately().

Options:
A. $$
\mathrm{20 nm}
$$
B. $$
\mathrm{25 nm}
$$
C. $$
\mathrm{5 n m}
$$
D. $$
\mathrm{3 mm}
$$
E. $$
\mathrm{10 \mu m}
$$
F. $$
\mathrm{1 mm}
$$
G. $$
\mathrm{0.5 pm}
$$
H. $$
\mathrm{8 pm}
$$
I. $$
\mathrm{2 \mu m}
$$
J. $$
\mathrm{15 cm}
$$
Answer: C
If a particle's kinetic energy becomes negative within a certain region of space, that region is classically forbidden. It is known that the total energy of a ground state hydrogen atom is -13.6 eV. Determine the classically forbidden region for the electron in this atom (i.e., the region where the electron's kinetic energy becomes negative) is.

Options:
A. $$
r \! > \! 0. 0 5 3 \mathrm{n m}
$$
B. $$
r \! > \! 0. 2 0 0 \mathrm{n m}
$$
C. $$
r \! > \! 0. 1 5 0 \mathrm{n m}
$$
D. $$
r \! > \! 0. 1 0 5 \mathrm{n m}
$$
E. $$
r \! > \! 0. 1 0 6 \mathrm{n m}
$$
F. $$
r \! < \! 0. 2 0 0 \mathrm{n m}
$$
G. $$
r \! < \! 0. 0 5 3 \mathrm{n m}
$$
H. $$
r \! < \! 0. 1 0 6 \mathrm{n m}
$$
I. $$
r \! < \! 0. 1 5 0 \mathrm{n m}
$$
J. $$
r \! < \! 0. 1 0 7 \mathrm{n m}
$$
Answer: E
The muon is one of the relatively long-lived particles, with a mass 207 times that of the electron, and its charge and other interaction properties are exactly the same as those of the electron. When an electron in a neutral atom is replaced by a $\mu^{-}$, it forms a $\mu$-atom.
What is the binding energy of the ground state of a $\mu$-hydrogen atom?

Options:
A. $$
1. 9 2 \times1 0^{3} ( \mathrm{e V} )
$$
B. $$
1. 8 4 \times1 0^{3} ( \mathrm{e V} )
$$
C. $$
4. 7 8 \times1 0^{2} ( \mathrm{e V} )
$$
D. $$
3. 6 5 \times1 0^{3} ( \mathrm{e V} )
$$
E. $$
2. 1 9 \times1 0^{3} ( \mathrm{e V} )
$$
F. $$
3. 2 7 \times1 0^{3} ( \mathrm{e V} )
$$
G. $$
2. 7 4 \times1 0^{3} ( \mathrm{e V} )
$$
H. $$
0. 8 9 \times1 0^{3} ( \mathrm{e V} )
$$
I. $$
2. 5 3 \times1 0^{3} ( \mathrm{e V} )
$$
J. $$
1. 5 3 \times1 0^{3} ( \mathrm{e V} )
$$
Answer: I
The hyperfine splitting of the ground state of the hydrogen atom is approximately () .

Options:
A. $$
7 0^{-5} \mathrm{e V}
$$
B. $$
6 0^{-8} \mathrm{e V}
$$
C. $$
3 0^{-7} \mathrm{e V}
$$
D. $$
2 0^{-6} \mathrm{e V}
$$
E. $$
9 0^{-7} \mathrm{e V}
$$
F. $$
1 0^{-8} \mathrm{e V}
$$
G. $$
8 0^{-9} \mathrm{e V}
$$
H. $$
5 0^{-9} \mathrm{e V}
$$
I. $$
1 0^{-7} \mathrm{e V}
$$
J. $$
4 0^{-7} \mathrm{e V}
$$
Answer: I
If an electron is confined within the dimensions of a nucleus, its momentum uncertainty is ().

Options:
A. $2~ \mathrm{MeV/c}$
B. $200~ \mathrm{MeV/c}$
C. $0.2~ \mathrm{keV/c}$
D. $20~ \mathrm{MeV/c}$
E. $2~ \mathrm{eV/c}$
F. $0. 2 ~\mathrm{eV/c}$
G. $0.02~ \mathrm{MeV/c}$
H. $20~ \mathrm{keV/c}$
I. $200~ \mathrm{keV/c}$
J. $2000~ \mathrm{eV/c}$
Answer: B
An electron is confined in a one-dimensional box and is in the ground state, with the box width $1 \times 10^{-10} \mathrm{m}$, and the electron energy is 38 eV.
Calculate the average force exerted on the walls of the box when the electron is in the ground state.

Options:
A. $$
2. 8 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
B. $$
6. 7 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
C. $$
0. 4 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
D. $$
7. 6 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
E. $$
5. 0 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
F. $$
3. 2 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
G. $$
8. 4 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
H. $$
9. 1 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
I. $$
4. 9 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
J. $$
1. 5 \times1 0^{9} \mathrm{e V} \cdot\mathrm{c m}^{-1}
$$
Answer: D
A non-relativistic particle with a mass of $m$ moves in one dimension in a potential $V (x)=-V_{0} \delta(x)$. When the particle is in the middle bind, determine the value of $x_{0}$ such that the probability of finding the particle within the region $| x | < x_{0}$ is 1/2.

Options:
A. $$
\frac{\hbar \sqrt{2}} {m V_{0}} $$
B. $$
\frac{\hbar^{2} \pi^{2}} {m V_{0}} $$
C. $$
\frac{\hbar^{3} \ln 2} {m V_{0}^{2}} $$
D. $$
\frac{2 \hbar^{2}} {m V_{0}} $$
E. $$
\frac{\hbar^{3} \ln 2} {m^{2} V_{0}^{2}} $$
F. $$
\frac{\hbar^{2} \ln 2} {2 m V_{0}} $$
G. $$
\frac{\hbar^{2} \pi} {2 m V_{0}} $$
H. $$
\frac{\hbar^{2} \operatorname{l n} 2} {m V_{0}}
$$
I. $$
\frac{\hbar^{2} \ln 3} {m V_{0}} $$
J. $$
\frac{\hbar^{2}} {2 m V_{0}} $$
Answer: H
A particle with mass $m$ is undergoing one-dimensional motion under the potential $V(x)$. Suppose it is in the energy eigenstate $\psi(x) = \left(\gamma^2 / \pi\right)^{1/4} \mathrm{e}^{-\gamma^2 x^2 / 2}$ with energy $E = \frac{\hbar^2 \gamma^2}{2m}$. Find the probability $P(p) \mathrm{d}p$ of the particle's momentum in the range $p \to p + \mathrm{d}p$.

Options:
A. $$
\sqrt{\frac{1} {\gamma \pi \hbar}} \mathrm{e}^{-\frac{p^{2}} {2 \gamma^{2} \hbar^{2}}} \mathrm{d} p
$$
B. $$
\sqrt{\frac{1} {\pi\gamma^{2} \hbar^{2}}} \mathrm{e}^{-\frac{p^{2}} {\gamma^{2} \hbar^{2}}} \mathrm{d} p
$$
C. $$
\sqrt{\frac{1} {\pi \hbar \gamma^{2}}} \mathrm{e}^{-\frac{(p - \gamma)^{2}} {\hbar^{2}}} \mathrm{d} p
$$
D. $$
\sqrt{\frac{1} {\pi \gamma^{2} \hbar^{3}}} \mathrm{e}^{-\frac{p^{2}} {\hbar^{2} \gamma}} \mathrm{d} p
$$
E. $$
\frac{1} {\sqrt{\pi \hbar} \gamma} \mathrm{e}^{-\frac{p^{2}} {\gamma^{2} \hbar}} \mathrm{d} p
$$
F. $$
\frac{1} {\sqrt{\pi} \gamma^{2} \hbar} \mathrm{e}^{-\frac{p^{2}} {\gamma \hbar^{2}}} \mathrm{d} p
$$
G. $$
\frac{1} {\gamma \hbar \sqrt{\pi}} \mathrm{e}^{-\frac{p^{2}} {\gamma^{3} \hbar^{2}}} \mathrm{d} p
$$
H. $$
\frac{1} {\gamma \sqrt{\pi \hbar^{2}}} \mathrm{e}^{-\frac{(p + \gamma)^{2}} {2 \hbar^{2}}} \mathrm{d} p
$$
I. $$
\sqrt{\frac{2} {\pi \hbar \gamma^{3}}} \mathrm{e}^{-\frac{p^{2}} {\gamma^{3} \hbar^{2}}} \mathrm{d} p
$$
J. $$
\sqrt{\frac{2} {\pi \gamma^{4} \hbar^{2}}} \mathrm{e}^{-\frac{p^{2}} {2 \gamma^{2} \hbar^{2}}} \mathrm{d} p
$$
Answer: B
Two electrons are tightly confined to different neighboring positions in a certain solid. In this way, they are distinguishable and can be described by their respective Pauli spin matrices $\sigma^{(1)}$ and $\sigma^{(2)}$. Their Hamiltonian is given by
$$
H = -J \left[ \sigma_{x}^{(1)} \sigma_{x}^{(2)} + \sigma_{y}^{(1)} \sigma_{y}^{(2)} \right]
$$
where $J$ is a constant.
How many energy levels does the system have? And what are the energies respectively?

Options:
A. $$
2
$$
$$
3 J, \ -3 J
$$
B. $$
3
$$
$$
2 J, \ 4 J, \ 6 J
$$
C. $$
3
$$
$$
0, \ -2 J, \ -4 J
$$
D. $$
2
$$
$$
4 J, \ -4 J
$$
E. $$
3
$$
$$
3 J, \ 0, \ -3 J
$$
F. $$
3
$$
$$
J, \ -J, \ -2 J
$$
G. $$
2
$$
$$
0, \ 8 J
$$
H. $$
3
$$
$$
2 J, \ 0, \ -2 J
$$
I. $$
3
$$
$$
4 J, \ 1 J, \ -4 J
$$
J. $$
2
$$
$$
6 J, \ -6 J
$$
Answer: H
The de Broglie wavelength of electrons accelerated by a $10,000 \mathrm{V}$ voltage is().

Options:
A. $$
0. 3 1^ {}\AA
$$
B. $$
1. 2 0^ {}\AA
$$
C. $$
1. 0 9^ {}\AA
$$
D. $$
1. 1 5^ {}\AA
$$
E. $$
0. 6 4^ {}\AA
$$
F. $$
2. 1 0^ {}\AA
$$
G. $$
0. 5 6^ {}\AA
$$
H. $$
2. 0 1^ {}\AA
$$
I. $$
0. 1 2^ {}\AA
$$
J. $$
0. 2 3^ {}\AA
$$
Answer: I
Assume that for an ideal gas, the ratio of $C_{p}$ to $C_{v}$, denoted as $\gamma$, is a function of temperature. Attempt to determine the relationship between $T$ and $V$ during a quasi-static process. This relationship involves a function $F(T)$ is_____, expressed as
$$
\operatorname{ln} F(T) = \int{\frac{\mathrm{d}T}{(\gamma-1)T}}
$$

Options:
A. $$
FT (V)=C
$$
B. $$
VF ( T)=C
$$
C. $$
CV (T)=F
$$
D. $$
TV (C)=F
$$
E. $$
VC (F)=T
$$
F. $$
VT (F)=C
$$
G. $$
TF (V)=C
$$
H. $$
CT (F)=V
$$
I. $$
FV (T)=C
$$
J. $$
FV (C)=T
$$
Answer: B
One mole of an ideal gas undergoes a reversible isothermal expansion process from a volume of $V_{1}$ to $2V_{1}$. What's the total change in entropy of the whole system?

Options:
A. $$RIn10$$
B. $$RIn5$$
C. $$RIn6$$
D. $$RIn4$$
E. $$RIn3$$
F. $$RIn11$$
G. $$0$$
H. $$RIn8$$
I. $$RIn9$$
J. $$RIn7$$
Answer: G
Conduction electrons in a metal can be regarded as a free electron gas. Given $N = 10^{22}/\mathbf{cm}^{3}$, what is the chemical potential of conduction electrons at $T=0 \mathrm{K}$ ?

Options:
A. $$
1. 7 ( \mathbf{e V} )
$$
B. $$
4. 8 ( \mathbf{e V} )
$$
C. $$
8. 9 ( \mathbf{e V} )
$$
D. $$
10. 0 ( \mathbf{e V} )
$$
E. $$
7. 3 ( \mathbf{e V} )
$$
F. $$
2. 5 ( \mathbf{e V} )
$$
G. $$
6. 4 ( \mathbf{e V} )
$$
H. $$
3. 2 ( \mathbf{e V} )
$$
I. $$
5. 6 ( \mathbf{e V} )
$$
J. $$
9. 1 ( \mathbf{e V} )
$$
Answer: A
Relativistic electron gas. Consider a degenerate (i.e., \( T=0 \mathrm{K} \)) gas consisting of \( N \) non-interacting electrons in a volume \( V \). Derive the equation relating the pressure, energy, and volume of this gas in the extreme relativistic case is _________.(Neglecting electron mass).

Options:
A. $$
P V={\frac{5} {6}} E
$$
B. $$
P V={\frac{1} {2}} E
$$
C. $$
P V={\frac{3} {2}} E
$$
D. $$
P V=2 E
$$
E. $$
P V=3 E
$$
F. $$
P V={\frac{2} {3}} E
$$
G. $$
P V=E
$$
H. $$
P V={\frac{1} {5}} E
$$
I. $$
P V={\frac{1} {3}} E
$$
J. $$
P V={\frac{1} {4}} E
$$
Answer: I
A stream of water is projected upward at an angle of 30°. To reach a wall located at a horizontal distance of 60 feet and a height of 10 feet, what is the minimum initial velocity required? Disregard friction.

Options:
A. $V_{0} \,=\, 7 0. \, 0 \mathrm{f t} / \mathrm{s e c}$
B. $V_{0} \,=\, 5 6. \, 0 \mathrm{f t} / \mathrm{s e c}$
C. $V_{0} \,=\, 5 8. \, 0 \mathrm{f t} / \mathrm{s e c}$
D. $V_{0} \,=\, 4 5. \, 0 \mathrm{f t} / \mathrm{s e c}$
E. $V_{0} \,=\, 6 0. \, 0 \mathrm{f t} / \mathrm{s e c}$
F. $V_{0} \,=\, 6 2. \, 0 \mathrm{f t} / \mathrm{s e c}$
G. $V_{0} \,=\, 5 9. \, 0 \mathrm{f t} / \mathrm{s e c}$
H. $V_{0} \,=\, 6 5. \, 0 \mathrm{f t} / \mathrm{s e c}$
I. $V_{0} \,=\, 5 1. \, 0 \mathrm{f t} / \mathrm{s e c}$
J. $V_{0} \,=\, 4 8. \, 0 \mathrm{f t} / \mathrm{s e c}$
Answer: B
The normal shock wave in the airflow occurs at $P_{1}=10 \mathrm{Pa}$ (70Pa abs), $T_{1}=40 \mathrm{~ °F ~} \ (5 \mathrm{~ °C ~})$, $V_{1}= 1400 \mathrm{ft/sec \left(425 \ m/s \right)}$, what is the temperature $T_{2}$ ?

Options:
A. $$
5 5 \mathrm{^\circ C}
$$
B. $$
2 0 \mathrm{^\circ C}
$$
C. $$
4 0 \mathrm{^\circ C}
$$
D. $$
9 0 \mathrm{^\circ C}
$$
E. $$
8 0 \mathrm{^\circ C}
$$
F. $$
1 0 \mathrm{^\circ C}
$$
G. $$
6 5 \mathrm{^\circ C}
$$
H. $$
2 5 \mathrm{^\circ C}
$$
I. $$
3 5 \mathrm{^\circ C}
$$
J. $$
7 0 \mathrm{^\circ C}
$$
Answer: A
A $\mathbf{p}$-type semiconductor sample measuring 30 mm in length, 6 mm in width, and 1 mm in thickness, has a resistance of $500\Omega$. When placed in a 0.5 T magnetic field perpendicular to the sample plane and a current of 10 mA passes through the sample, a Hall voltage of $5 \mathrm{mV}$ appears. What is the Hall mobility of the semiconductor sample?

Options:
A. $$
9 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
B. $$
2 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
C. $$
6 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
D. $$
1 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
E. $$
7 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
F. $$
8 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
G. $$
1 5^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
H. $$
5 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
I. $$
4 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
J. $$
3 0^{2} [ \mathrm{\ c m^{2} ( \ V \cdot\ s )} ]
$$
Answer: D
Given that the impurity concentration gradient $\alpha_{\mathrm{j}}$ of a silicon linear graded junction is $10^{22} / \mathrm{cm}^{4}$, and $V_{\mathrm{D}}=0.68 \mathrm{V}$. What is the value of $x_{\mathrm{D}}$ when a 10V reverse bias is applied?

Options:
A. $$
6. 6 1 \times1 0^{-9} ( \mathrm{\, c m} )
$$
B. $$
4. 3 9 \times1 0^{-5} ( \mathrm{\, c m} )
$$
C. $$
5. 0 0 \times1 0^{-6} ( \mathrm{\, c m} )
$$
D. $$
7. 3 3 \times1 0^{-4} ( \mathrm{\, c m} )
$$
E. $$
9. 4 5 \times1 0^{-4} ( \mathrm{\, c m} )
$$
F. $$
5. 7 6 \times1 0^{-7} ( \mathrm{\, c m} )
$$
G. $$
2. 1 5 \times1 0^{-6} ( \mathrm{\, c m} )
$$
H. $$
1. 9 8 \times1 0^{-2} ( \mathrm{\, c m} )
$$
I. $$
3. 8 2 \times1 0^{-8} ( \mathrm{\, c m} )
$$
J. $$
8. 2 0 \times1 0^{-3} ( \mathrm{\, c m} )
$$
Answer: B
The electrical conductivity of intrinsic germanium is known to be $3.56 \times 10^{-2} \, \mathrm{S/cm}$ at 310K and $0.42 \times 10^{-2} \, \mathrm{S/cm}$ at 273K. For an n-type germanium sample at these two temperatures, the donor impurity concentration $N_{\mathrm{D}} = 10^{15} \, \mathrm{cm}^{-3}$. What is the conductivity of doped germanium at the aforementioned temperatures, given that the electron mobility $\mu_{\mathrm{n}} = 3600 \, \mathrm{cm^2/(V \cdot s)}$ and the hole mobility $\mu_{\mathrm{p}} = 1700 \, \mathrm{cm^2/(V \cdot s)}$?

Options:
A. $$
9. 3 1 {\times} 1 0^{-3} ( \mathrm{S / c m} )
$$
B. $$
5. 9 4 {\times} 1 0^{-3} ( \mathrm{S / c m} )
$$
C. $$
1. 3 6 {\times} 1 0^{-2} ( \mathrm{S / c m} )
$$
D. $$
8. 0 5 {\times} 1 0^{-2} ( \mathrm{S / c m} )
$$
E. $$
5. 7 6 {\times} 1 0^{-1} ( \mathrm{S / c m} )
$$
F. $$
6. 8 5 {\times} 1 0^{-2} ( \mathrm{S / c m} )
$$
G. $$
3. 5 0 {\times} 1 0^{-2} ( \mathrm{S / c m} )
$$
H. $$
2. 1 7 {\times} 1 0^{-1} ( \mathrm{S / c m} )
$$
I. $$
7. 4 8 {\times} 1 0^{-2} ( \mathrm{S / c m} )
$$
J. $$
4. 2 9 {\times} 1 0^{-1} ( \mathrm{S / c m} )
$$
Answer: E