MikaStars39/nano-eval huggingface dataset:
A rectangular waveguide has dimensions of $a \times b = 23 \, \mathrm{mm} \times 10 \, \mathrm{mm}$, filled with an ideal dielectric with $\varepsilon_{\tau} = 9$. The waveguide transmits a $\mathrm{TE}_{10}$ mode at an operating frequency of 3GHz. Calculate $\lambda_{g}$, $v_{\mathrm{p}}$, and $Z_{\mathrm{TE}_{10}}$.

Options:
A. $$
4 6. 8 9 \ \mathrm{m m}
$$
$$
1. 4 2 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 0. 9 3 \ \Omega
$$
B. $$
4 9. 5 0 \ \mathrm{m m}
$$
$$
1. 3 8 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 4. 1 2 \ \Omega
$$
C. $$
4 9. 1 5 \ \mathrm{m m}
$$
$$
1. 3 9 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 3. 5 6 \ \Omega
$$
D. $$
4 8. 3 7 \ \mathrm{m m}
$$
$$
1. 4 5 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 2. 3 6 \ \Omega
$$
E. $$
4 6. 3 2 \ \mathrm{m m}
$$
$$
1. 3 7 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 1. 9 8 \ \Omega
$$
F. $$
5 0. 7 8 \ \mathrm{m m}
$$
$$
1. 5 0 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 9 2. 4 7 \ \Omega
$$
G. $$
5 0. 2 4 \ \mathrm{m m}
$$
$$
1. 3 6 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 7 9. 5 2 \ \Omega
$$
H. $$
4 7. 1 8 \ \mathrm{m m}
$$
$$
1. 4 0 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 0. 2 1 \ \Omega
$$
I. $$
5 1. 3 4 \ \mathrm{m m}
$$
$$
1. 5 1 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 9 0. 6 3 \ \Omega
$$
J. $$
4 8. 9 9 \ \mathrm{m m}
$$
$$
1. 4 8 \times1 0^{8} \mathrm{~ m / s}
$$
$$
1 8 5. 7 4 \ \Omega
$$
Answer: D
The cross-sectional dimensions of a known air-filled rectangular waveguide are $a \times b = 23 \text{ mm} \times 10 \text{ mm}$, where two magnetic field components of the $\mathrm{TM}_{11}$ mode are transmitted.
$$
\dot{H}_{x}=\frac{j \omega\varepsilon} {k_{c}^{2}} \ \frac{\pi} {b} \dot{E}_{0} \operatorname{s i n} \frac{\pi x} {a} \ \operatorname{c o s} \frac{\pi y} {b} \mathrm{e}^{-j\beta z}
$$
$$
\dot{H}_{y}=-\frac{\mathrm{j} \omega\varepsilon} {k_{c}^{2}} \frac{\pi} {a} \dot{E}_{0} \cos\frac{\pi x} {a} \sin\frac{\pi y} {b} \mathrm{e}^{-\mathrm{j} \beta x}
$$
When the signal source frequency $f_{0}$ is twice the cutoff frequency $f_{c}$, determine the propagation constant $\gamma$, the guided wavelength $\lambda_{\mathbf{g}}$, and the wave impedance $Z_{\mathrm{T M}}$ for this mode.

Options:
A. $$
\gamma=\mathrm{j} 6 2 0. 0 0 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{\mathbf{g}}=1. 0 2 5 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 4 0. \ 5 5 \quad{\Omega}
$$
B. $$
\gamma=\mathrm{j} 5 7 0. 4 9 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{\mathbf{g}}=1. 1 0 0 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 2 0. \ 6 8 \quad{\Omega}
$$
C. $$
\gamma=\mathrm{j} 6 0 5. 2 7 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{\mathbf{g}}=1. 1 5 0 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 0 0. \ 0 0 \quad {\Omega}
$$
D. $$
\gamma==\mathrm{j} 5 9 3. 3 9 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{g}=1. 0 3 9 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=2 9 9. \ 1 3 \quad{\Omega}
$$
E. $$
\gamma=\mathrm{j} 5 6 7. 8 8 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{g}=1. 0 9 5 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 1 2. \ 9 9 \quad{\Omega}
$$
F. $$
\gamma=\mathrm{j} 5 8 0. 1 8 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{\mathbf{g}}=1. 0 7 5 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 3 5. \ 2 2 \quad {\Omega}
$$
G. $$
\gamma==\mathrm{j} 6 0 1. 1 9 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{s}=1. 0 1 5 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 5 5. \ 7 5 \quad{\Omega}
$$
H. $$
\gamma=\mathrm{j} 5 9 3. 3 7 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{g}=1. 1 2 0 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 4 6. \ 3 3 \quad{\Omega}
$$
I. $$
\gamma==\mathrm{j} 5 9 3. 3 9 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{s}=1. 0 5 9 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 2 6. \ 4 9 \quad {\Omega}
$$
J. $$
\gamma==\mathrm{j} 6 0 5. 7 5 \quad\mathrm{r a d} / \mathrm{m}
$$
$$
\lambda_{s}=1. 0 9 0 \quad\mathrm{c m}
$$
$$
Z_{\mathrm{T M}_{1 1}}=3 2 0. \ 0 6 \quad{\Omega}
$$
Answer: I
The radiation power of a certain antenna is 50 W, and the average power density in the far field at a distance of 1 km in its maximum radiation direction is 0.5 mW/m². If the radiation power remains unchanged and the original antenna is replaced with an elementary dipole, how many times is the average power density of the original antenna at 1 km in the maximum radiation direction compared to that of the elementary dipole?

Options:
A. 92.02
B. 67.03
C. 80.04
D. 88.07
E. 82.06
F. 90.01
G. 95.08
H. 76.05
I. 70.09
J. 84.03
Answer: J
Given that the relative dielectric constant of the microstrip line's substrate is $\varepsilon_{r}=3.78$, the thickness is $h=0.6 \ \mathrm{mm}$, and the conductor width is $W=1 \ \mathrm{mm}$. What are the effective dielectric constant and characteristic impedance of this microstrip line?

Options:
A. $$
2. 4 6 8
$$
$$
6 9. 7 5 \Omega
$$
B. $$
2. 8 7 5
$$
$$
6 1. 2 4 \Omega
$$
C. $$
4. 1 0 5
$$
$$
7 0. 4 1 \Omega
$$
D. $$
3. 7 3 2
$$
$$
5 3. 9 7 \Omega
$$
E. $$
2. 6 5 4x
$$
$$
5 9. 8 2 \Omega
$$
F. $$
3. 0 8 6
$$
$$
6 4. 1 9 \Omega
$$
G. $$
3. 5 9 0
$$
$$
6 5. 2 0 \Omega
$$
H. $$
3. 2 7 9
$$
$$
6 6. 0 2 \Omega
$$
I. $$
2. 9 1 5
$$
$$
6 7. 3 8 \Omega
$$
J. $$
3. 1 2 0
$$
$$
5 5. 8 3 \Omega
$$
Answer: B
A known rectangular resonant cavity filled with air, with dimensions $a \times b \times l = 2.5 \ \mathrm{cm} \times 1.25 \ \mathrm{cm} \times 6 \ \mathrm{cm}$, resonates in the $\mathrm{TE}_{102}$ mode. If the cavity is filled with a dielectric medium and resonates at the same resonant frequency in the $\mathrm{TE}_{103}$ mode, find the relative permittivity of the dielectric medium.

Options:
A. 3.18
B. 0.84
C. 1.51
D. 2.31
E. 0.97
F. 1.23
G. 0.68
H. 1.99
I. 2.87
J. 2.45
Answer: C
Design a $- 3 \lambda_{0} / 4$ type coaxial resonant cavity, with $\lambda_{0} = 5 \ \mathrm{cm}$. To reduce the cavity length, a capacitive load is used. If the length of the inner conductor is shortened by 0.8 cm, what is the value of the loading capacitance?

Options:
A. $$
7 6. 7 \Omega
$$
B. $$
5 4. 3 \Omega
$$
C. $$
7 2. 4 \Omega
$$
D. $$
8 0. 0 \Omega
$$
E. $$
8 1. 5 \Omega
$$
F. $$
7 3. 1 \Omega
$$
G. $$
6 9. 2 \Omega
$$
H. $$
6 7. 9 \Omega
$$
I. $$
6 6. 5 \Omega
$$
J. $$
5 8. 6 \Omega
$$
Answer: A
A rectangular waveguide of a standing wave measurement line has dimensions of $a \times b = 2.54 \ \mathrm{cm} \times 1.27 \ \mathrm{cm}$, and the distance measured between two adjacent wave nodes is 3 cm. When the end of the measurement line is connected to an antenna, the measured standing wave ratio $\rho$ is 3, and the distance from the first wave node in the waveguide to the antenna is measured to be 1.3 cm. Calculate the operating frequency in the waveguide.

Options:
A. $9.54GHz$
B. $5.62GHz$
C. $3.55GHz$
D. $10.23GHz$
E. $7.73GHz$
F. $8.19GHz$
G. $11.07GHz$
H. $6.88GHz$
I. $4.87GHz$
J. $6.45GHz$
Answer: E
The spacing between a seven-element Yagi antenna's reflector and the driven element is 0.15λ. Each director and the spacing between them and the driven element are arranged at equal intervals, with a spacing of 0.2λ. Try to estimate its directivity factor and main lobe width.

Options:
A. $$
D=8.47
$$
$$
2_{\alpha_{0. 5}}=45.6^{\circ}
$$
B. $$
D=10.93
$$
$$
2_{\alpha_{0. 5}}=51.3^{\circ}
$$
C. $$
D=13.02
$$
$$
2_{\alpha_{0. 5}}=58.5^{\circ}
$$
D. $$
D=7.76
$$
$$
2_{\alpha_{0. 5}}=52.4^{\circ}
$$
E. $$
D=12.05
$$
$$
2_{\alpha_{0. 5}}=60.8^{\circ}
$$
F. $$
D=9.21
$$
$$
2_{\alpha_{0. 5}}=39.7^{\circ}
$$
G. $$
D=11.32
$$
$$
2_{\alpha_{0. 5}}=54.0^{\circ}
$$
H. $$
D=11.89
$$
$$
2_{\alpha_{0. 5}}=47.9^{\circ}
$$
I. $$
D=10.50
$$
$$
2_{\alpha_{0. 5}}=44.1^{\circ}
$$
J. $$
D=9.88
$$
$$
2_{\alpha_{0. 5}}=50.2^{\circ}
$$
Answer: B
Two infinitely large parallel conducting planes are separated by a distance of $d$. At point $P$, which is a distance $a$ from the surface of one of the conductors, there is a point charge with a charge amount of $q$. Given that the potentials of both conductors are zero, calculate the induced charge amounts on the surfaces of the two conducting planes.

Options:
A. $$
Q_2 = \frac{a}{d} q\\
Q_1 = \frac{d-a}{d} q\\
$$
B. $$
Q_2 = \frac{d-a}{a} q\\
Q_1 = -\frac{a}{d} q\\
$$
C. $$
Q_2 = -\frac{a}{d} q\\
Q_1 = -\frac{a}{d-a} q\\
$$
D. $$
Q_2 = -\frac{d-a}{d} q\\
Q_1 = -\frac{a}{d} q\\
$$
E. $$
Q_2 = \frac{a}{d-a} q\\
Q_1 = -\frac{d-a}{d} q\\
$$
F. $$
Q_2 = -\frac{a}{d} q\\
Q_1 = -\frac{d-a}{d} q\\
$$
G. $$
Q_2 = -\frac{a}{d} q\\
Q_1 = \frac{a}{d} q\\
$$
H. $$
Q_2 = \frac{d-a}{a} q\\
Q_1 = -\frac{d-a}{d} q\\
$$
I. $$
Q_2 = -\frac{d}{a} q\\
Q_1 = \frac{d-a}{d} q\\
$$
J. $$
Q_2 = \frac{d-a}{d} q\\
Q_1 = -\frac{a}{d-a} q\\
$$
Answer: F
In a cloud chamber experiment, a neutral $\Delta$ particle with an energy of 10 GeV was observed. It subsequently decayed into a proton and a negative $\pi$ meson. It is known that the rest mass of the $\Lambda$ particle is $M c^{2} = 1115 \mathrm{MeV}$, and its lifetime is $\tau = 2.9 \times 10^{-10}$ s. The rest mass of the proton is $m_{1} c^{2} \approx 939 \mathrm{MeV}$, and the rest mass of the $\pi$ meson is $m_{2} c^{2} \approx 140 \mathrm{MeV}$. (1) What is the average distance traveled by the $\Lambda$ particle in the cloud chamber before it decays?

Options:
A. 0.32(m)
B. 0.89(m)
C. 1.15(m)
D. 0.45(m)
E. 0.54(m)
F. 0.78(m)
G. 1.30(m)
H. 1.00(m)
I. 1.50(m)
J. 0.68(m)
Answer: F
There are two parallel rods, each with a length of $l_{0}$ in their respective rest frames. They are moving at the same speed $\mathbf{v}$ relative to a certain reference frame, but in opposite directions and parallel to the rods. Find the length of one rod as measured by an observer standing on the other rod.

Options:
A. $l = l_0 \left(1 - \frac{v^2}{c^2}\right)$
B. $l = \frac{l_0}{\sqrt{1 + \frac{v^2}{c^2}}}$
C. $l = \frac{l_0}{\sqrt{1 - \frac{v^2}{c^2}}}$
D. $l = l_0 \sqrt{1 - \frac{v^2}{c^2}}$
E. $l = l_0 \left(\frac{v^2}{c^2} - 1\right)$
F. $l = l_0 \sqrt{1 + \frac{v^2}{c^2}}$
G. $l = l_0 \left(\frac{v}{c}\right)^2$
H. $l = \frac{l_0}{2} \sqrt{1 - \frac{v^2}{c^2}}$
I. $l = 2l_0 \sqrt{1 - \frac{v^2}{c^2}}$
J. $l = l_0 \sqrt{\frac{v^2}{c^2} - 1}$
Answer: D
The mean lifetime of an unstable atomic nucleus is $4.0 \times 10^{-6}$ seconds. It undergoes decay when moving relative to the laboratory at a velocity of 0.60 $c$ (where $c$ is the speed of light in a vacuum), emitting an electron. The speed of the electron relative to it is 0.90 $c$; in the reference frame that follows its motion, the velocity of the electron is perpendicular to its direction of motion. Given that the rest mass of the electron is $m$, find its mean lifetime.

Options:
A. $\tau = 3.5 \times 10^{-6} \, \text{s}$
B. $\tau = 7.5 \times 10^{-6} \, \text{s}$
C. $\tau = 4.0 \times 10^{-6} \, \text{s}$
D. $\tau = 6.0 \times 10^{-6} \, \text{s}$
E. $\tau = 8.0 \times 10^{-6} \, \text{s}$
F. $\tau = 7.0 \times 10^{-6} \, \text{s}$
G. $\tau = 5.5 \times 10^{-6} \, \text{s}$
H. $\tau = 4.5 \times 10^{-6} \, \text{s}$
I. $\tau = 5.0 \times 10^{-6} \, \text{s}$
J. $\tau = 6.5 \times 10^{-6} \, \text{s}$
Answer: I
Based on past production data, the defective rate of a certain product is 2%. If a 95% confidence interval is required, and the estimation error is to be no more than 4%, how large a sample should be taken?

Options:
A. $$
n
\approx7 5
$$
B. $$
n
\approx2 4
$$
C. $$
n
\approx5 0
$$
D. $$
n
\approx6 2
$$
E. $$
n
\approx9 3
$$
F. $$
n
\approx8 1
$$
G. $$
n
\approx4 8
$$
H. $$
n
\approx1 0
$$
I. $$
n
\approx3 9
$$
J. $$
n
\approx1 6
$$
Answer: G
A certain product's trademark is "MAXAM", but two letters have fallen off. Someone picks them up and randomly puts them back. What is the probability that it still reads "MAXAM" after they are put back?

Options:
A. 0.3
B. 0.4
C. 0.6
D. 0.0
E. 0.7
F. 0.9
G. 0.1
H. 0.8
I. 0.5
J. 0.2
Answer: C
Assume the lifetime of a certain electronic device (measured in hours) $T$ follows an exponential distribution with the probability density
$$
f(t) = \left\{\begin{matrix} \lambda e^{-\mu} & {t > 0,} \\ 0 & {t \leq 0,} \end{matrix} \right.
$$
where $\lambda > 0$ is unknown. From this batch of devices, randomly select $n$ units to undergo an independent lifetime test starting at time $t=0$. The test continues until the predetermined time $T_{0}$. At this point, $k$ devices ($0 < k < n$) have failed. Calculate the maximum likelihood estimate of the failure rate.

Options:
A. $$
\hat{\lambda} \!=\! {\frac{1} {n}} \operatorname{l n} {\frac{n \!+\! k} {T_{0}}}
$$
B. $$
\hat{\lambda} \!=\! {\frac{k} {n}} T_{0} \operatorname{l n} {\frac{n \!+\! k} {n}}
$$
C. $$
\hat{\lambda} \!=\! {\frac{1} {T_{0}}} \operatorname{l n} {\frac{n} {n \!-\! k}}
$$
D. $$
\hat{\lambda} \!=\! {\frac{1} {T_{0}}} \operatorname{l n} {\frac{n \!-\! k} {k}}
$$
E. $$
\hat{\lambda} \!=\! {\frac{1} {T_{0}}} \operatorname{l n} {\frac{k} {n \!-\! k}}
$$
F. $$
\hat{\lambda} \!=\! {\frac{k} {T_{0}}} \operatorname{l n} {\frac{n} {k}}
$$
G. $$
\hat{\lambda} \!=\! {\frac{k} {T_{0}}} \operatorname{l n} {\frac{n \!-\! k} {n}}
$$
H. $$
\hat{\lambda} \!=\! {\frac{k} {n}} \operatorname{l n} {\frac{T_{0}} {k \!+\! 1}}
$$
I. $$
\hat{\lambda} \!=\! {\frac{1} {n}} \operatorname{l n} {\frac{T_{0}} {n \!+\! k}}
$$
J. $$
\hat{\lambda} \!=\! {\frac{n} {k}} \operatorname{l n} {\frac{k \!-\! 1} {T_{0}}}
$$
Answer: C
At 25 °C and 101,325 Pa, what is the collision frequency between $\mathrm{N_{2}}$ molecules and $\mathrm{O_{2}}$ molecules in the air? What is the average time for each collision?

Options:
A. $$
1. \, 2 9 5 \, \times\, 1 0^{3 2} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
8 \, \times\, 1 0^{-3 4} \mathrm{~ s}
$$
B. $$
3. \, 9 4 0 \, \times\, 1 0^{3 3} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
3 \, \times\, 1 0^{-3 5} \mathrm{~ s}
$$
C. $$
2. \, 3 8 0 \, \times\, 1 0^{3 2} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
5 \, ×\, 1 0^{-3 6} \mathrm{~ s}
$$
D. $$
4. \, 3 5 7 \, \times\, 1 0^{3 4} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
6 \, \times\, 1 0^{-3 6} \mathrm{~ s}
$$
E. $$
2. \, 7 1 8 \, \times\, 1 0^{3 4} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
9 \, \times\, 1 0^{-3 6} \mathrm{~ s}
$$
F. $$
4. \, 1 4 5 \, \times\, 1 0^{3 5} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
1 \, \times\, 1 0^{-3 4} \mathrm{~ s}
$$
G. $$
2. \, 6 0 0 \, \times\, 1 0^{3 5} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
2 \, \times\, 1 0^{-3 5} \mathrm{~ s}
$$
H. $$
1. \, 8 5 0 \, \times\, 1 0^{3 3} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
5 \, \times\, 1 0^{-3 5} \mathrm{~ s}
$$
I. $$
3. \, 1 0 2 \, \times\, 1 0^{3 2} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
7 \, \times\, 1 0^{-3 4} \mathrm{~ s}
$$
J. $$
2. \, 4 9 5 \, \times\, 1 0^{3 4} \mathrm{~ m^{-3} ~} \cdot\mathrm{~ s^{-1} ~},
$$
$$
4 \, \times\, 1 0^{-3 5} \mathrm{~ s}
$$
Answer: J
Keep pressure at standard pressure to calculate the standard molar entropy of acetone vapor at $1000 \mathrm{K}$. It is known that the standard molar entropy of acetone vapor at $298 \mathrm{K}$ is $S_{\mathrm{m}}^{\mathrm{e}}(298 \, \mathrm{K}) = 294.9 \, \mathrm{J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1}$. Within the temperature range of $273 \sim 1500 \mathrm{~K}$, the molar heat capacity at constant pressure of acetone vapor, $C_{p, m}^{0}$, is related to temperature by the expression:
$$
C_{p, m}^{0} = [ 22.47 + 201.8 \times 10^{-3} \, (T / K) - 63.5 \times 10^{-6} \, (T / K)^{2} ] \; \mathrm{J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1}
$$

Options:
A. $$
3 5 0. 7 6 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
B. $$
3 8 2. 1 7 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
C. $$
3 9 0. 2 5 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
D. $$
4 3 4. 8 3 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
E. $$
4 1 5. 3 2 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
F. $$
4 2 5. 6 4 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
G. $$
4 0 0. 5 8 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
H. $$
3 6 8. 4 4 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
I. $$
3 8 9. 3 0 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
J. $$
4 5 0. 1 9 \mathrm{~ J ~ \cdot~ m o l^{-1} ~ \cdot~ K^{-1} ~}
$$
Answer: D
At 293 K, in ammonia aqueous solution A, the ratio of $\mathrm{NH}_{3}$ to $\mathrm{H}_{2}\mathrm{O}$ is 1:8.5, and the partial pressure of $\mathrm{NH}_{3}$ above solution A is 10.64 kPa. In ammonia aqueous solution B, the ratio of $\mathrm{NH}_{3}$ to $\mathrm{H}_{2}\mathrm{O}$ is 1:21, and the partial pressure of $\mathrm{NH}_{3}$ above solution B is 3.597 kPa. Calculate the $\Delta G$ when transferring 1 mol of $\mathrm{NH}_{3} (\text{g})$ from a large amount of solution A to a large amount of solution B at the same temperature.

Options:
A. $$
- 2 \; 3 9 9 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
B. $$
- 3 \; 0 5 1 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
C. $$
- 1 \; 9 8 3 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
D. $$
- 2 \; 4 7 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
E. $$
- 3 \; 1 6 7 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
F. $$
- 2 \; 6 4 2 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
G. $$
- 3 \; 4 2 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
H. $$
- 2 \; 8 7 3 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
I. $$
- 2 \; 5 1 7 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
J. $$
- 2 \; 0 2 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
Answer: F
The decomposition reaction of $\mathrm{PCl}_5(\mathrm{g})$ is $\mathrm{PCl}_5(\mathrm{g}) \rightleftharpoons \mathrm{PCl}_3(\mathrm{g}) + \mathrm{Cl}_2(\mathrm{g})$. At 523 K and 100 kPa, equilibrium is achieved, and the density of the equilibrium mixture is measured to be $\rho = 2.695 ~ \mathrm{kg} \cdot \mathrm{m}^{-3}$. Calculate the degree of dissociation of $\mathrm{PCl_5(\mathrm{g})}$.

Options:
A. $$
\alpha=1. 0 0 0
$$
B. $$
\alpha=0. 1 1 1
$$
C. $$
\alpha=0. 8 8 8
$$
D. $$
\alpha=0. 6 6 6
$$
E. $$
\alpha=0. 2 2 2
$$
F. $$
\alpha=0. 4 4 4
$$
G. $$
\alpha=0. 9 9 9
$$
H. $$
\alpha=0. 7 7 7
$$
I. $$
\alpha=0. 3 3 3
$$
J. $$
\alpha=0. 5 5 5
$$
Answer: H
At 298 K, a KCl solution with a conductivity of $1.289 \mathrm{~S \cdot~ m^{-1} ~}$ is placed in a conductivity cell, and a resistance of 23.782 is measured. If a $2.414 \times 10^{-3} \, \mathrm{mol} \cdot \mathrm{dm}^{-3}$ HAc (acetic acid) solution is introduced into this conductivity cell, the measured resistance is 3942 Ω. Calculate the degree of dissociation and the dissociation equilibrium constant of this HAc solution.

Options:
A. $$0.090,
2. 0 1 \times1 0^{-6}
$$
B. $$0.081,
5. 3 1 \times1 0^{-8}
$$
C. $$0.078,
1. 5 4 \times1 0^{-4}
$$
D. $$0.105,
3. 5 8 \times1 0^{-4}
$$
E. $$0.045,
6. 8 5 \times1 0^{-5}
$$
F. $$0.099,
4. 2 0 \times1 0^{-7}
$$
G. $$0.073,
1. 0 9 \times1 0^{-5}
$$
H. $$0.056,
2. 5 3 \times1 0^{-6}
$$
I. $$0.082,
1. 7 7 \times1 0^{-5}
$$
J. $$0.064,
3. 6 3 \times1 0^{-7}
$$
Answer: I
At 273.15 K and 101,325 Pa, 1 dm³ of water can dissolve 49 mol of oxygen or 23.5 mol of nitrogen. Under standard conditions, the amount of air that can dissolve in 1 dm³ of water is _____.

Options:
A. $$
4 2. 7 \mathrm{\ m o l}
$$
B. $$
2 8. 6 \mathrm{\ m o l}
$$
C. $$
1 8. 5 \mathrm{\ m o l}
$$
D. $$
4 8. 1 \mathrm{\ m o l}
$$
E. $$
5 0. 3 \mathrm{\ m o l}
$$
F. $$
1 9. 8 \mathrm{\ m o l}
$$
G. $$
2 5. 4 \mathrm{\ m o l}
$$
H. $$
3 7. 4 \mathrm{\ m o l}
$$
I. $$
3 4. 9 \mathrm{\ m o l}
$$
J. $$
1 5. 2 \mathrm{\ m o l}
$$
Answer: B
The main repulsive energy in quantum solids is the zero-point energy of the atoms. Consider a rough one-dimensional model of the crystal $\mathbf{He}^{4}$, where each He atom is confined within a segment of length $L$. The wave function of a free particle with a half-wavelength is taken as the ground state wave function. The zero-point kinetic energy for each particle is ().

Options:
A. $$
\frac{\hbar^{2}} {8 L m^{2}}
$$
B. $$
\frac{\hbar^{2} L} {8 m}
$$
C. $$
\frac{\hbar^{2}} {16 m L^{2}}
$$
D. $$
\frac{\hbar L^{2}} {8 m}
$$
E. $$
\frac{\hbar^{2}} {2 m L^{2}}
$$
F. $$
\frac{2 \hbar^{2}} {8 m L^{2}}
$$
G. $$
\frac{\hbar^{2}} {8 m L}
$$
H. $$
\frac{\hbar^{2}} {8 m L^{2}}
$$
I. $$
\frac{\hbar L} {8 m}
$$
J. $$
\frac{\hbar^{2}} {4 m L^{2}}
$$
Answer: H
The root mean square (RMS) value of the thermal expansion $\frac{\Delta V}{V}$ of the sodium unit cell at 300K is estimatedly (). The bulk modulus is taken as $7 \times 10^{10} \ \mathrm{erg} \ \mathrm{cm}^{-3}$. Note that the Debye temperature of 158K is smaller than 300K, so the thermal energy is on the order of $\mathbf{K}_{\mathrm{B}} \mathbf{T}$.

Options:
A. $$
\frac{\Delta V} {V} \approx0. 1 2 5
$$
B. $$
\frac{\Delta V} {V} \approx0. 0 5 6
$$
C. $$
\frac{\Delta V} {V} \approx0. 6 7 9
$$
D. $$
\frac{\Delta V} {V} \approx0. 0 8 4
$$
E. $$
\frac{\Delta V} {V} \approx0. 2 5 7
$$
F. $$
\frac{\Delta V} {V} \approx0. 4 2 1
$$
G. $$
\frac{\Delta V} {V} \approx0. 1 0 2
$$
H. $$
\frac{\Delta V} {V} \approx0. 3 3 3
$$
I. $$
\frac{\Delta V} {V} \approx0. 0 3 0
$$
J. $$
\frac{\Delta V} {V} \approx0. 0 9 8
$$
Answer: I
Indium antimonide has a bandgap $E_{g} = 0.23 \text{ ev}$; dielectric constant $\varepsilon = 18$; and electron effective mass $m_{t} = 0.015 m$. The ionization energy of the donor is ().

Options:
A. $$
2. 6 4 \times1 0^{-3} \mathbf{e v}
$$
B. $$
4. 4 7 \times1 0^{-3} \mathbf{e v}
$$
C. $$
1. 8 3 \times1 0^{-4} \mathbf{e v}
$$
D. $$
7. 9 5 \times1 0^{-4} \mathbf{e v}
$$
E. $$
6. 5 6 \times1 0^{-3} \mathbf{e v}
$$
F. $$
6. 2 8 \times1 0^{-4} \mathbf{e v}
$$
G. $$
9. 1 1 \times1 0^{-4} \mathbf{e v}
$$
H. $$
8. 0 0 \times1 0^{-4} \mathbf{e v}
$$
I. $$
5. 0 2 \times1 0^{-4} \mathbf{e v}
$$
J. $$
3. 1 4 \times1 0^{-3} \mathbf{e v}
$$
Answer: F
Derive the expression for the binding energy of an electron in a one-dimensional single square potential well, which is (), with a potential well depth of $U_{0}$ and width $a$. Assume the solution is symmetric about the midpoint of the potential well.

Options:
A. $$
v^{2}-u^{2}=\frac{m a^{2} U_{0}} {24\hbar^{1}} \,.
$$
B. $$
u^{2}-v^{2}=\frac{n a^{2} U_{0}} {4 \hbar^{1}} \,.
$$
C. $$
v^{2}-u^{2}=\frac{m a^{2} U_{0}} {2 \hbar^{1}} \,.
$$
D. $$
u^{2}+v^{2}=\frac{m a^{2} V_{0}} {4 \hbar^{1}} \,.
$$
E. $$
u^{3}+v^{3}=\frac{m a^{2} U_{0}} {4 \hbar^{1}} \,.
$$
F. $$
u^{2}+v^{2}=\frac{m a^{2} U_{0}} {2 \hbar^{1}} \,.
$$
G. $$
u^{2}+v^{2}=\frac{m a^{2} U_{0}} {3 \hbar^{1}} \,.
$$
H. $$
u^{3}+v^{3}=\frac{m a^{3} U_{0}} {2 \hbar^{1}} \,.
$$
I. $$
u^{2}-v^{2}=\frac{m a^{2} U_{0}} {2 \hbar^{1}} \,.
$$
Answer: F
Assuming the energy required to extract a sodium atom from the body and place it on the surface is $1ev$, calculate the concentration of Schottky vacancies at 300K, which is ().

Options:
A. $$
7. 5 \times1 0^{-1 8}
$$
B. $$
2. 9 \times1 0^{-1 8}
$$
C. $$
1. 6 \times1 0^{-1 7}
$$
D. $$
8. 4 \times1 0^{-1 6}
$$
E. $$
4. 8 \times1 0^{-1 9}
$$
F. $$
3. 7 \times1 0^{-1 7}
$$
G. $$
3. 2 \times1 0^{-1 6}
$$
H. $$
5. 1 \times1 0^{-1 5}
$$
I. $$
6. 0 \times1 0^{-1 9}
$$
J. $$
9. 3 \times1 0^{-1 5}
$$
Answer: C
Calculating the bonding energy ratio of Ne in body-centered cubic and face-centered cubic structures using the Lennard-Jones Potential, which is ().

Options:
A. 1.203
B. 0.676
C. 1.065
D. 1.362
E. 0.842
F. 0.983
G. 0.957
H. 1.431
I. 0.748
J. 1.119
Answer: G
In the one-dimensional lattice, use $| n \rangle$ to represent the $s$-state of the $n$-th lattice point. Under the tight-binding approximation, which considers only nearest-neighbor interactions, write the expression for the matrix element $\langle m | \hat{H} | n \rangle$.

Options:
A. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}+J_{0} \delta_{m, n}+J_{1} \delta_{m, n \mp1}
$$
B. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{g}} \delta_{m, n}+J_{0} \delta_{m, n}-J_{2} \delta_{m, n \pm1}
$$
C. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{f}} \delta_{m, n}-J_{3} \delta_{m, n}-J_{1} \delta_{m, n \mp1}
$$
D. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{p}} \delta_{m, n}+J_{0} \delta_{m, n}-J_{1} \delta_{m, n \pm1}
$$
E. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}+J_{0} \delta_{m, n}-J_{1} \delta_{m, n \pm3}
$$
F. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}+J_{1} \delta_{m, n}-J_{0} \delta_{m, n \pm2}
$$
G. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}-J_{0} \delta_{m, n}+J_{2} \delta_{m, n \pm2}
$$
H. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}-J_{4} \delta_{m, n}-J_{0} \delta_{m, n \pm1}
$$
I. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{s}} \delta_{m, n}-J_{0} \delta_{m, n}-J_{1} \delta_{m, n \pm1}
$$
J. $$
\langle m | \hat{H} | n \rangle=\varepsilon_{\mathrm{d}} \delta_{m, n}-J_{2} \delta_{m, n}+J_{1} \delta_{m, n \pm1}
$$
Answer: I
It is known that $\mathrm{Si}$ contains only donor impurities with $N_{\mathrm{D}}=10^{15} \, \mathrm{cm^{-3}}$. At 40K, the measured electron concentration is $10^{12} \, \mathrm{cm}^{-3}$. Estimate the ionization energy of the donor impurities.

Options:
A. $$
\approx0. 0 8 9 \, 0 \, \mathrm{e V}
$$
B. $$
\approx0. 1 0 4 \, 5 \, \mathrm{e V}
$$
C. $$
\approx0. 1 2 3 \, 4 \, \mathrm{e V}
$$
D. $$
\approx0. 0 3 4 \, 7 \, \mathrm{e V}
$$
E. $$
\approx0. 0 6 7 \, 3 \, \mathrm{e V}
$$
F. $$
\approx0. 0 7 2 \, 2 \, \mathrm{e V}
$$
G. $$
\approx0. 0 8 2 \, 0 \, \mathrm{e V}
$$
H. $$
\approx0. 0 9 5 \, 8 \, \mathrm{e V}
$$
I. $$
\approx0. 0 5 6 \, 1 \, \mathrm{e V}
$$
J. $$
\approx0. 0 2 3 \, 8 \, \mathrm{e V}
$$
Answer: F
A balloon with a diameter of 0.3 m is filled with air at a pressure of 150 kPa. As a result of heating, the balloon's diameter reversibly increases to 0.4 m. Given that the air pressure changes proportionally with the balloon's diameter, calculate the work done by the air in this process

Options:
A. $$
3. 1 4 \times1 0^{4} J
$$
B. $$
4. 4 5 \times1 0^{3} J
$$
C. $$
1. 2 5 \times1 0^{4} J
$$
D. $$
3. 9 9 \times1 0^{4} J
$$
E. $$
2. 6 2 \times1 0^{3} J
$$
F. $$
2. 3 7 \times1 0^{4} J
$$
G. $$
1. 6 6 \times1 0^{4} J
$$
H. $$
2. 8 8 \times1 0^{3} J
$$
I. $$
5. 1 5 \times1 0^{4} J
$$
J. $$
4. 2 7 \times1 0^{4} J
$$
Answer: G
8 kg of gas expands in a plastic container from 1 MPa and 1 m³ to 0.5 MPa. The change in the thermodynamic energy of the gas during the process is -40 kJ/kg, with \( p v^{1.2} = \) constant. Determine the amount of heat exchanged between the gas and the surrounding environment

Options:
A. $$
2 3 1. 8 k J
$$
B. $$
1 7 5. 5 k J
$$
C. $$
2 5 0. 9 k J
$$
D. $$
1 8 0. 3 k J
$$
E. $$
2 9 0. 4 k J
$$
F. $$
2 0 5. 7 k J
$$
G. $$
2 6 0. 1 k J
$$
H. $$
2 2 4. 6 k J
$$
I. $$
1 9 5. 2 k J
$$
J. $$
2 7 4. 6 k J
$$
Answer: H
A rigid air storage tank with a volume of 0.027 m³ contains air at a pressure of $7 \times 10^{5}Pa$ and a temperature of$20^{\circ} \mathrm{C}$. The tank is equipped with an exhaust valve that opens when the pressure reaches \($8.75 \times 10^{5}$\) $Pa$ and closes when the pressure drops to \($8.4 \times 10^{5}$\) $Pa$. If the valve opens due to external heating, how much air is lost due to heating? Assume the air temperature inside the tank remains constant during the exhaust process.

Options:
A. $$
0. 0 1 8 k g
$$
B. $$
0. 0 0 9 k g
$$
C. $$
0. 0 1 5 k g
$$
D. $$
0. 0 0 8 k g
$$
E. $$
0. 0 0 5 k g
$$
F. $$
0. 0 2 0 k g
$$
G. $$
0. 0 0 7 k g
$$
H. $$
0. 0 0 6 k g
$$
I. $$
0. 0 1 2 k g
$$
J. $$
0. 0 1 0 k g
$$
Answer: B
The state of air entering the piston compressor is $p_{1}=0.1 \; \mathrm{MPa}, t_{1}=-27^{\circ} \ \mathrm{C}$. It is compressed through a reversible adiabatic process to $t_{2}=207^{\circ} \; \mathrm{C}$. Determine the final pressure of the air after compression, the change in thermodynamic energy, and the amount of work exchanged between the air and the external environment during the process

Options:
A. $$
0. 6 3 1 M P a
$$
$$
1 5 4. 1 9 k J / k g
$$
$$
- 1 4 3 k J / k g
$$
B. $$
0. 5 7 1 M P a
$$
$$
1 3 8. 5 2 k J / k g
$$
$$
- 1 2 8 k J / k g
$$
C. $$
0. 5 1 8 M P a
$$
$$
1 2 9. 0 6 k J / k g
$$
$$
- 130.72 k J / k g
$$
D. $$
0. 4 8 2 M P a
$$
$$
1 0 3. 8 4 k J / k g
$$
$$
- 1 0 5 k J / k g
$$
E. $$
0. 5 6 0 M P a
$$
$$
1 3 5. 6 1 k J / k g
$$
$$
- 1 3 0 k J / k g
$$
F. $$
0. 4 9 5 M P a
$$
$$
1 1 7. 2 5 k J / k g
$$
$$
- 1 2 4 k J / k g
$$
G. $$
0. 5 0 7 M P a
$$
$$
1 2 1. 6 8 k J / k g
$$
$$
- 1 1 9 k J / k g
$$
H. $$
0. 5 4 4 M P a
$$
$$
1 2 7. 9 8 k J / k g
$$
$$
- 1 2 6 k J / k g
$$
I. $$
0. 5 8 2 M P a
$$
$$
1 4 2. 4 9 k J / k g
$$
$$
- 1 3 5 k J / k g
$$
J. $$
0. 5 3 2 M P a
$$
$$
1 4 0. 0 3 k J / k g
$$
$$
- 1 3 1 k J / k g
$$
Answer: C
For a certain converging nozzle, the outlet cross-sectional area is 5 cm². At the inlet cross-section, the air parameters are \( $p_{1}$ = $0.6 \ \text{MPa}$ \) and \( $t_{1}$ = $580 \ \degree \text{C}$ \). What is the back pressure when the maximum mass flow rate \( $q_{m,\text{max}}$ \) can be achieved? What is \( $q_{m,\text{max}}$ \)?

Options:
A. $$
0. 5 0 \times1 0^{5} P a
$$
$$
0. 3 9 k g / s
$$
B. $$
0. 4 5 \times1 0^{5} P a
$$
$$
0. 3 6 k g / s
$$
C. $$
0. 3 8 \times1 0^{5} P a
$$
$$
0. 4 5 k g / s
$$
D. $$
0. 3 2 \times1 0^{5} P a
$$
$$
0. 4 2 k g / s
$$
E. $$
0. 3 4 \times1 0^{5} P a
$$
$$
0. 4 1 k g / s
$$
F. $$
0. 4 0 \times1 0^{5} P a
$$
$$
0. 3 7 k g / s
$$
G. $$
0. 2 5 \times1 0^{5} P a
$$
$$
0. 5 0 k g / s
$$
H. $$
0. 2 8 \times1 0^{5} P a
$$
$$
0. 4 8 k g / s
$$
I. $$
0. 6 0 \times1 0^{5} P a
$$
$$
0. 4 0 k g / s
$$
J. $$
0. 7 0 \times1 0^{5} P a
$$
$$
0. 3 8 k g / s
$$
Answer: D
The pressure of the steam in a certain condenser is known to be 5 kPa, with a specific volume \( $v$ = $25.38 \, \mathrm{m}^3/\mathrm{kg}$ \). Determine the state of the steam and the values of \($t$ \), \( $h$ \), and \( $s$ \)

Options:
A. $$
2 7. 2 4 1 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 1 6 4. 7 \mathrm{~ k J / k g}
$$
$$
7. 1 7 8 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
B. $$
3 3. 9 5 4 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 3 3 2. 6 \mathrm{~ k J / k g}
$$
$$
7. 7 9 9 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
C. $$
3 4. 0 1 7 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 4 1 2. 3 \mathrm{~ k J / k g}
$$
$$
8. 3 5 0 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
D. $$
2 8. 6 8 3 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 1 9 5. 2 \mathrm{~ k J / k g}
$$
$$
6. 7 3 6 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
E. $$
3 5. 3 8 9 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 4 7 2. 1 \mathrm{~ k J / k g}
$$
$$
8. 2 3 9 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
F. $$
3 6. 2 7 4 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 5 0 1. 4 \mathrm{~ k J / k g}
$$
$$
8. 5 6 4 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
G. $$
2 9. 6 3 5 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 1 8 0. 8 \mathrm{~ k J / k g}
$$
$$
6. 8 2 7 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
H. $$
3 1. 5 9 8 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 2 5 7. 9 \mathrm{~ k J / k g}
$$
$$
7. 5 4 1 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
I. $$
3 2. 8 7 9 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 3 1 8. 3 \mathrm{~ k J / k g}
$$
$$
7. 6 0 1 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
J. $$
3 0. 4 5 2 \ 3^ {\circ} {\mathsf{C}}
$$
$$
2 0 8 0. 5 \mathrm{~ k J / k g}
$$
$$
6. 9 8 7 \ 3 \ \mathrm{k J} / ( \mathrm{k g " K} )
$$
Answer: I
A certain steam power device uses a single reheat cycle, with the steam turbine inlet pressure at $17 MPa$ and a temperature of $540^{\circ} \ \mathrm{C}$. Reheating occurs when it expands to $4 MPa$, with the reheater exit temperature at $540^{\circ} \ \mathrm{C}$, and the exhaust pressure is $0.008 MPa$. Determine the cycle thermal efficiency.

Options:
A. 0.442
B. 0.498
C. 0.448
D. 0.467
E. 0.412
F. 0.441
G. 0.472
H. 0.443
I. 0.495
J. 0.423
Answer: C
A certain compression refrigeration equipment uses ammonia as the refrigerant. The evaporation temperature of ammonia is known to be $-10^{\circ} \ \mathrm{C}$, and the condensation temperature is 38 $\mathrm{c}$. The compressor intake is dry saturated ammonia vapor, and the required refrigeration capacity is $10^8 \mathrm{~kJ/h}$. Calculate the refrigerant flow rate, the compressor power consumption, and the coefficient of performance (COP)

Options:
A. $$
1 \ 0 3 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 5 0 \ \mathrm{k W}
$$
$$
5. 0
$$
B. $$
1 \ 1 2 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
5. 9 5 \ \mathrm{k W}
$$
$$
3. 9
$$
C. $$
1 \ 1 5 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 9 0 \ \mathrm{k W}
$$
$$
4. 4
$$
D. $$
1 \ 1 0 5 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 3 0 \ \mathrm{k W}
$$
$$
4. 3
$$
E. $$
1 \ 0 7 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 4 0 \ \mathrm{k W}
$$
$$
3. 7
$$
F. $$
9 \ 8 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
7. 5 0 \ \mathrm{k W}
$$
$$
4. 1
$$
G. $$
9 \ 5 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
7. 0 0 \ \mathrm{k W}
$$
$$
4. 2
$$
H. $$
1 \ 0 8 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 1 7 \ \mathrm{k W}
$$
$$
4. 5
$$
I. $$
1 \ 0 0 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
6. 7 5 \ \mathrm{k W}
$$
$$
4. 0
$$
J. $$
1 \ 2 5 0 \ \ \mathrm{k J} / \mathrm{k g}
$$
$$
5. 3 5 \ \mathrm{k W}
$$
$$
3. 8
$$
Answer: H
A large steel plate with a thickness of 5 cm and an initial temperature of 250°C is suddenly placed in an airstream at a temperature of 20°C. The surface heat transfer coefficient of convective heat exchange between the steel plate surface and the airstream is $100 \, \mathrm{W/(m^{2} \cdot K)}$. Given that the thermal conductivity of the steel plate is $\lambda = 47 \, \mathrm{W/(m \cdot K)}$ and the thermal diffusivity is $a = 1.47 \times 10^{-5} \, \mathrm{m^{2}/s}$, find the time required for the surface temperature of the steel plate to reach 150°C

Options:
A. 9.8min
B. 3.2min
C. 6.9min
D. 11.0min
E. 5.4min
F. 8.7min
G. 15.3min
H. 7.6min
I. 12.5min
J. 4.5min
Answer: H
On an autumn night, the sky is clear, and the outdoor air temperature is 2°C. The cosmic background radiation temperature is approximately 3 K. There is a steel plate facing space, with insulation underneath. If the convective heat transfer coefficient between the plate and the air is 10 W/(m²·K). Calculate the thermal equilibrium temperature of the steel plate

Options:
A. $$
5. 8^{\circ} \mathrm{C}
$$
B. $$
-4. 5^{\circ} \mathrm{C}
$$
C. $$
1 5. 5^{\circ} \mathrm{C}
$$
D. $$
-2 5. 3^{\circ} \mathrm{C}
$$
E. $$
9. 1^{\circ} \mathrm{C}
$$
F. $$
-2 0. 9^{\circ} \mathrm{C}
$$
G. $$
0. 4^{\circ} \mathrm{C}
$$
H. $$
2 3. 4^{\circ} \mathrm{C}
$$
I. $$
-3 2. 7^{\circ} \mathrm{C}
$$
J. $$
-1. 4^{\circ} \mathrm{C}
$$
Answer: F
In a thermal power plant, there is a superheated steam pipeline (steel pipe) with an outer diameter of \($100 \, \mathrm{mm}$\), insulated with glass wool having a thermal conductivity of \($\lambda=0.04 \, \mathrm{W/(m \cdot K)}$\). It is known that the temperature of the outer surface of the steel pipe is \($400^{\circ} \mathrm{C}$\). The outer surface temperature of the insulation layer must not exceed \($50^{\circ} \mathrm{C}$\), and the heat loss per meter of pipeline must be less than \($160 \, \mathrm{W}$\). Determine the thickness of the insulation layer

Options:
A. 29.75mm
B. 48.05mm
C. 26.45mm
D. 33.50mm
E. 23.90mm
F. 44.80mm
G. 36.65mm
H. 39.20mm
I. 45.60mm
J. 41.30mm
Answer: G
According to the crystal's ball model, if the diameter of the sphere remains unchanged, how much does the volume expand when $\mathrm{Fe}$ transforms from face-centered cubic (fcc) to body-centered cubic (bcc)?

Options:
A. 11%
B. 6%
C. 10%
D. 12%
E. 9%
F. 14%
G. 8%
H. 7%
I. 15%
J. 5%
Answer: E
What is the density of the diamond structure?

Options:
A. 1.19
B. 1.08
C. 0.78
D. 0.41
E. 0.92
F. 0.27
G. 1.52
H. 0.53
I. 0.34
J. 0.66
Answer: I
The crystal structure of Nb is body-centered cubic (bcc), with a lattice constant of $0.3294 \, \text{nm}$ and a density of $8.57 \, \text{g/cm}^3$. What is the number of vacancies in every $10^6$ Nb?

Options:
A. 6895.2
B. 7779.1
C. 8032.5
D. 7634.7
E. 7176.6
F. 7412.8
G. 6921.3
H. 7581.4
I. 7149.0
J. 7248.5
Answer: E
At $20^\circ \mathrm{C}$, $\mathbf{W}$ has one vacancy in every $10^{23}$ lattice sites. When the temperature is raised from 20°C to 1020°C, the lattice constant expands by $(4 \times 10^{-4})$ 1%, and the density decreases by 0.012%. What are the vacancy formation energy and formation entropy of $\mathbf{W}$?

Options:
A. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 8 2 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=2. 9 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
B. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 9 3 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=2. 3 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
C. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 2 2 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=9. 9 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
D. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 4 8 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=7. 4 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
E. $$
\begin{array} {l} {{E_{\mathrm{v}}=0. 9 7 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=6. 1 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
F. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 6 5 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=2. 5 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
G. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 3 0 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=5. 6 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
H. $$
\begin{array} {l} {{E_{\mathrm{v}}=2. 0 1 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=4. 2 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
I. $$
\begin{array} {l} {{E_{\mathrm{v}}=1. 4 5 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=3. 3 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
J. $$
\begin{array} {l} {{E_{\mathrm{v}}=2. 5 0 ( \mathrm{e V} )}} \\ {{S_{\mathrm{v}}=1. 8 \times1 0^{-4} ( \mathrm{e V} )}} \\ \end{array}
$$
Answer: I
A piece of carbon steel with $w ({\bf C})=0.1$ %is carburized at 930°C. At a depth of 0.05 cm, the carbon concentration reaches 0.45%. For all times $t > 0$, the carburizing atmosphere maintains a surface composition of 1%. Assuming $D_{\mathrm{c}}^{\mathrm{\gamma}}=2. \ 0 \times1 0^{-5}$ $\operatorname{e x p} (-1 4 0 0 0 0 / R T ) \left( {\bf m}^{2} / {\bf s} \right)$ , what is the carburizing time?

Options:
A. $$
t \approx1. 4 \times1 0^{-1} ( {\bf s} )
$$
B. $$
t \approx2. 0 \times1 0^{5} ( {\bf s} )
$$
C. $$
t \approx1. 8 \times1 0^{8} ( {\bf s} )
$$
D. $$
t \approx2. 5 \times1 0^{6} ( {\bf s} )
$$
E. $$
t \approx2. 1 \times1 0^{2} ( {\bf s} )
$$
F. $$
t \approx1. 7 \times1 0^{-2} ( {\bf s} )
$$
G. $$
t \approx1. 5 \times1 0^{3} ( {\bf s} )
$$
H. $$
t \approx1. 9 \times1 0^{-3} ( {\bf s} )
$$
I. $$
t \approx1. 0 \times1 0^{4} ( {\bf s} )
$$
J. $$
t \approx1. 3 \times1 0^{7} ( {\bf s} )
$$
Answer: I
Prepare a single crystal aluminum tensile test specimen (with a cross-sectional area of $9mm^{2}$) for wide-temperature tensile testing. The tensile axis intersects at an angle of 36.7° with [001], 19.1° with [011], and 22.2° with [111]. The load at the onset of yielding is $20.4N$. What is the resolved shear stress of the primary slip system?

Options:
A. $$
1. 2 5 ( {\bf M P a} )
$$
B. $$
2. 1 9 ( {\bf M P a} )
$$
C. $$
0. 8 7 ( {\bf M P a} )
$$
D. $$
1. 0 1 ( {\bf M P a} )
$$
E. $$
1. 5 3 ( {\bf M P a} )
$$
F. $$
0. 6 5 ( {\bf M P a} )
$$
G. $$
1. 7 4 ( {\bf M P a} )
$$
H. $$
1. 3 0 ( {\bf M P a} )
$$
I. $$
0. 9 2 ( {\bf M P a} )
$$
J. $$
1. 4 6 ( {\bf M P a} )
$$
Answer: D
After cold working, the dislocation density of $\mathrm{Ag}$ is $10^{12} / \mathrm{cm}^{2}$. Assuming that the newly recrystallized nuclei move towards the deformed matrix with a large angle grain boundary, calculate the minimum curvature radius of the grain boundary bowing out is_____. $\mathrm{(Ag, G=30 ~ GPa, b=0.3 ~ nm, \gamma=0.4 ~ J/m^{2})}$.

Options:
A. 30(nm)
B. 31(nm)
C. 29(nm)
D. 32(nm)
E. 27(nm)
F. 35(nm)
G. 26(nm)
H. 24(nm)
I. 28(nm)
J. 25(nm)
Answer: C
The eutectic reaction in the Mg-Ni system is as follows:
$$
\mathrm{L} ( w ( \mathrm{Ni} )=23.57\% ) \rightarrow \mathrm{a} (\mathrm{Ni})+\mathrm{Mg}_{3} \mathrm{Ni} [ w ( \mathrm{Ni} )=54.67\% ]
$$
Let $\mathrm{C}_{1}$ be the hypoeutectic alloy and $\mathrm{C}_{t}$ the hypereutectic alloy. The mass fractions of the proeutectic phase in these two alloys are equal, but the total amount of α in the $\mathbf{C}_{1}$ alloy is 2.5 times that in the $\mathbf{C}_{t}$ alloy. What are the compositions of $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$?

Options:
A. $C_1=13.1\%$
$C_2=36.7\%$
B. $C_1=11.1\%$
$C_2=39.5\%$
C. $C_1=15.0\%$
$C_2=33.2\%$
D. $C_1=17.4\%$
$C_2=30.9\%$
E. $C_1=14.9\%$
$C_2=34.8\%$
F. $C_1=16.5\%$
$C_2=32.1\%$
G. $C_1=14.3\%$
$C_2=35.6\%$
H. $C_1=10.2\%$
$C_2=40.4\%$
I. $C_1=11.8\%$
$C_2=38.9\%$
J. $C_1=12.7\%$
$C_2=37.8\%$
Answer: J
The deformed material undergoes isothermal annealing, and after recrystallization is completed, it is kept at a constant temperature to allow the grain size to grow. When annealed for 30 minutes, the measured grain diameter of the material is 23 μm, and the corresponding yield strength is 112 MPa; when annealed for 60 minutes, the measured yield strength is 103 MPa. Determine the yield strength of the material when annealed for 90 minutes is______.(Assume the time required for recrystallization and the grain size at the end of recrystallization can be neglected) (10 points)

Options:
A. $$
1 0 1 . 8 ( \mathrm{M P a} )
$$
B. $$
9 6 . 0 ( \mathrm{M P a} )
$$
C. $$
1 1 0 . 5 ( \mathrm{M P a} )
$$
D. $$
1 0 9 . 3 ( \mathrm{M P a} )
$$
E. $$
1 0 0 . 2 ( \mathrm{M P a} )
$$
F. $$
9 9 . 9 ( \mathrm{M P a} )
$$
G. $$
9 5 . 7 ( \mathrm{M P a} )
$$
H. $$
9 8. 4 ( \mathrm{M P a} )
$$
I. $$
1 0 2 . 6 ( \mathrm{M P a} )
$$
J. $$
1 0 7 . 1 ( \mathrm{M P a} )
$$
Answer: H
To achieve homogeneous nucleation at 2045K, what atmospheric pressure increase is required? Given the volume change upon solidification $\Delta V = -0.26 \ \mathrm{cm}^3 / \mathrm{mol}$ and (1) $= 9.87 \times 10^6 \ \mathrm{cm}^3 \cdot \mathrm{Pa}$.

Options:
A. $$
1 0 5 \, 4 5 0 \times1 0^{5} \, ( \mathrm{P a} )
$$
B. $$
1 1 6 \, 3 6 6 \times1 0^{5} \, ( \mathrm{P a} )
$$
C. $$
1 2 3 \, 4 4 4 \times1 0^{5} \, ( \mathrm{P a} )
$$
D. $$
1 4 0 \, 0 0 0 \times1 0^{5} \, ( \mathrm{P a} )
$$
E. $$
9 8 7 \, 2 0 0 \times1 0^{5} \, ( \mathrm{P a} )
$$
F. $$
1 3 4 \, 5 3 4 \times1 0^{5} \, ( \mathrm{P a} )
$$
G. $$
1 2 7 \, 5 0 0 \times1 0^{5} \, ( \mathrm{P a} )
$$
H. $$
9 9 9 \, 9 9 9 \times1 0^{5} \, ( \mathrm{P a} )
$$
I. $$
9 5 2 \, 1 1 5 \times1 0^{5} \, ( \mathrm{P a} )
$$
J. $$
1 0 3 \, 2 1 9 \times1 0^{5} \, ( \mathrm{P a} )
$$
Answer: B
If the energy of a photon is equal to the rest energy of an electron, what are the frequency, wavelength, and momentum of this photon?

Options:
A. $$
1. 1 9 3 4 \times1 0^{2 1} ( \mathrm{H z} )
$$
$$
2. 2 7 0 \times1 0^{-1 3} ( \mathrm{m} )
$$
$$
2. \, 8 4 7 5 \, \times\, 1 0^{-2 3} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
B. $$
1. 1 7 3 8 \times1 0^{2 1} ( \mathrm{H z} )
$$
$$
2. 3 1 5 \times1 0^{-1 0} ( \mathrm{m} )
$$
$$
2. \, 5 7 0 6 \, \times\, 1 0^{-2 1} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
C. $$
1. 1 8 7 5 \times1 0^{1 9} ( \mathrm{H z} )
$$
$$
2. 2 8 8 \times1 0^{-1 0} ( \mathrm{m} )
$$
$$
2. \, 6 0 9 2 \, \times\, 1 0^{-2 0} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
D. $$
1. 2 2 5 9 \times1 0^{2 0} ( \mathrm{H z} )
$$
$$
2. 3 0 6 \times1 0^{-1 3} ( \mathrm{m} )
$$
$$
2. \, 9 2 4 1 \, \times\, 1 0^{-2 5} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
E. $$
1. 2 8 1 6 \times1 0^{2 0} ( \mathrm{H z} )
$$
$$
2. 2 9 7 \times1 0^{-1 2} ( \mathrm{m} )
$$
$$
2. \, 7 1 8 4 \, \times\, 1 0^{-2 4} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
F. $$
1. 2 0 5 6 \times1 0^{1 9} ( \mathrm{H z} )
$$
$$
2. 2 4 5 \times1 0^{-1 1} ( \mathrm{m} )
$$
$$
2. \, 8 1 6 7 \, \times\, 1 0^{-2 4} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
G. $$
1. 1 9 6 4 \times1 0^{2 2} ( \mathrm{H z} )
$$
$$
2. 2 3 1 \times1 0^{-1 2} ( \mathrm{m} )
$$
$$
2. \, 9 3 8 2 \, \times\, 1 0^{-2 3} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
H. $$
1. 2 4 1 9 \times1 0^{1 8} ( \mathrm{H z} )
$$
$$
2. 3 1 0 \times1 0^{-1 1} ( \mathrm{m} )
$$
$$
2. \, 5 6 2 1 \, \times\, 1 0^{-2 1} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
I. $$
1. 2 7 4 3 \times1 0^{1 8} ( \mathrm{H z} )
$$
$$
2. 2 5 2 \times1 0^{-1 0} ( \mathrm{m} )
$$
$$
2. \, 6 7 5 4 \, \times\, 1 0^{-2 2} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
J. $$
1. 2 3 5 6 \times1 0^{2 0} ( \mathrm{H z} )
$$
$$
2. 2 6 4 \times1 0^{-1 2} ( \mathrm{m} )
$$
$$
2. \, 7 3 0 9 \, \times\, 1 0^{-2 2} ( \mathbf{k g} \cdot\mathbf{m} \cdot\mathbf{s}^{-1} )
$$
Answer: J
The emission wavelength of the excited state of the atom is assumed to be 0m. If the measurement accuracy of the wavelength is $\frac{\Delta\lambda} {\lambda}= 10^{-7}$, what is the average lifetime of this atomic state?

Options:
A. $$
\approx6. 4 \times1 0^{-3} ( \mathrm{s} )
$$
B. $$
\approx9. 1 \times1 0^{-2} ( \mathrm{s} )
$$
C. $$
\approx5. 6 \times1 0^{-8} ( \mathrm{s} )
$$
D. $$
\approx8. 3 \times1 0^{-5} ( \mathrm{s} )
$$
E. $$
\approx1. 4 \times1 0^{-1} ( \mathrm{s} )
$$
F. $$
\approx7. 9 \times1 0^{-4} ( \mathrm{s} )
$$
G. $$
\approx2. 5 \times1 0^{-6} ( \mathrm{s} )
$$
H. $$
\approx8. 7 \times1 0^{0} ( \mathrm{s} )
$$
I. $$
\approx3. 2 \times1 0^{-9} ( \mathrm{s} )
$$
J. $$
\approx4. 1 \times1 0^{-7} ( \mathrm{s} )
$$
Answer: I
Estimate the electric field strength required to pull an electron away from an atom during the time it approximately takes for the electron to orbit around the atomic nucleus once is().

Options:
A. $$
\approx7 \times1 0^{9} Z^{1.8} \mathrm{V / m}
$$
B. $$
\approx2 \times1 0^{4} Z^{3} \mathrm{V / m}
$$
C. $$
\approx5 \times1 0^{5} Z \mathrm{V / m}
$$
D. $$
\approx6 \times1 0^{2} Z \mathrm{V / m}
$$
E. $$
\approx1 \times1 0^{5} Z^{2.8} \mathrm{V / m}
$$
F. $$
\approx9 \times1 0^{3} Z^{2.5} \mathrm{V / m}
$$
G. $$
\approx8 \times1 0^{6} Z^{1.5} \mathrm{V / m}
$$
H. $$
\approx1 \times1 0^{8} Z^{3.2} \mathrm{V / m}
$$
I. $$
\approx3 \times1 0^{6} Z^{2} \mathrm{V / m}
$$
J. $$
\approx4 \times1 0^{7} Z^{3.5} \mathrm{V / m}
$$
Answer: B
The $\mu^{-}$ can be considered a "heavy electron," with a mass approximately 200 times that of an electron. If the ground state energy of the hydrogen atom is $E_{0}$, then the ground state energy of the exotic atom formed by the $\mu^{-}$ and the proton is approximately ().

Options:
A. $$
1 5 0 \ E_{0}
$$
B. $$
3 5 0 \ E_{0}
$$
C. $$
2 0 0 \ E_{0}
$$
D. $$
3 0 0 \ E_{0}
$$
E. $$
2 2 0 \ E_{0}
$$
F. $$
2 3 0 \ E_{0}
$$
G. $$
1 8 0 \ E_{0}
$$
H. $$
1 6 0 \ E_{0}
$$
I. $$
2 5 0 \ E_{0}
$$
J. $$
1 7 0 \ E_{0}
$$
Answer: C
The spin-orbit splitting of the hydrogen atom's $2 \mathbf{p}$ state is approximately ().

Options:
A. $$
7 0^{-7} \mathrm{e V}
$$
B. $$
1 0^{-3} \mathrm{e V}
$$
C. $$
9 0^{-9} \mathrm{e V}
$$
D. $$
5 0^{-3} \mathrm{e V}
$$
E. $$
3 0^{-2} \mathrm{e V}
$$
F. $$
8 0^{-5} \mathrm{e V}
$$
G. $$
1 0^{-4} \mathrm{e V}
$$
H. $$
2 0^{-1} \mathrm{e V}
$$
I. $$
4 0^{-8} \mathrm{e V}
$$
J. $$
6 0^{-6} \mathrm{e V}
$$
Answer: G
What is the wavelength of the photon emitted when the spin orientation changes, with the electron in a magnetic field of 0.5T?

Options:
A. $$
2. 4 5 ( \mathrm{c m} )
$$
B. $$
2. 5 6 ( \mathrm{c m} )
$$
C. $$
2. 1 4 ( \mathrm{c m} )
$$
D. $$
1. 8 6 ( \mathrm{c m} )
$$
E. $$
3. 1 2 ( \mathrm{c m} )
$$
F. $$
1. 6 7 ( \mathrm{c m} )
$$
G. $$
2. 3 0 ( \mathrm{c m} )
$$
H. $$
2. 9 3 ( \mathrm{c m} )
$$
I. $$
2. 0 8 ( \mathrm{c m} )
$$
J. $$
1. 7 4 ( \mathrm{c m} )
$$
Answer: C
The function $\psi(x, t)$ is a solution to the one-dimensional Schrödinger equation for a free particle with mass $m$. It is known that at the initial moment, the state is:
$$
\psi(x, 0) = A \mathrm{e}^{-x^{2} / a^{2}}
$$
Determine the probability amplitude in momentum space at $t=0$.

Options:
A. $$
\left( \frac{A a} {4 \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {2 \hbar^{2}}}
$$
B. $$
\left( \frac{a} {2 \hbar} \right)^{1 / 2} A \mathrm{e}^{-\frac{a^{2} p^{2}} {\hbar^{2}}}
$$
C. $$
\left( \frac{A a} {2 \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{p^{2} a^{2}} {2 \hbar^{2}}}
$$
D. $$
\left( \frac{A a} {\hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {4}}
$$
E. $$
\left( \frac{A a} {2 p^2 \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2}} {4 \hbar^{2}}}
$$
F. $$
\left( \frac{\hbar A a} {2} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {\hbar^{2}}}
$$
G. $$
\left( \frac{A} {2 a \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {4 \hbar}}
$$
H. $$
\left( \frac{A a^2} {2 \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a p^{2}} {4 \hbar}}
$$
I. $$
\left( \frac{2 A a} {\hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {8 \hbar^{2}}}
$$
J. $$
\left( \frac{A a} {2 \hbar} \right)^{1 / 2} \mathrm{e}^{-\frac{a^{2} p^{2}} {4 \hbar^{2}}}
$$
Answer: J