MikaStars39/nano-eval huggingface dataset:
A G-M refrigerator operating between high and low pressures of 1.0 MPa and 0.1 MPa, using helium as the working fluid, has an ambient temperature of 300 K. The gas temperature before expansion is 80 K, and the expansion process is a reversible adiabatic compression process. Assume the efficiency of the compressor and regenerator is 100%. Determine the specific cooling capacity and the coefficient of performance.

Options:
A. $$7 0 5 3. 6 2 W / k g,2.2360$$
B. $$8 6 0 7. 7 4 W / k g,1.4142$$
C. $$2 3 9 9. 4 1 W / k g,0.5772$$
D. $$4 2 7 1. 2 3 W / k g,1.6180$$
E. $$5 7 8 9. 1 5 W / k g,0.6931$$
F. $$9 4 7 2. 5 6 W / k g,2.7183$$
G. $$1 8 4 6. 9 1 W / k g,3.1415$$
H. $$6 1 2 5. 8 4 W / k g,1.7321$$
I. $$3 5 8 8. 3 7 W / k g,3.1412$$
J. $$3 3 4 4. 5 9 W / k g,3.3333$$
Answer: I
The cascade refrigeration system consists of two-stage compression using R-22 and single-stage compression using R-23. The condensing temperature of the high-temperature section is $35^{\circ} \mathrm{C}$, the evaporating temperature of the low-temperature section is $-86^{\circ} \mathrm{C}$, and the condensing temperature of the low-temperature section is $-50^{\circ} \mathrm{C}$. To improve the operation of the refrigeration compressor in the low-temperature section, a Gas-liquid recuperator is added to the R-23 refrigeration system. This raises the low-pressure refrigerant vapor temperature by $\Delta t_{0 - 1}=30 \, \mathrm{°C}$ to subcool the high-pressure refrigerant liquid. To enhance the overall cycle efficiency and reduce the thermal load on the condenser-evaporator, a water-cooled cooler is used to cool the exhaust temperature $t_{3}$ of the low-temperature section compressor to $t_{3^{\prime}}=40 \mathrm{°C}$. Additionally, a gas-gas heat exchanger is installed to further increase the low-pressure refrigerant vapor temperature by $\Delta t_{1 - 2}=30 \mathrm{°C}$, which is used to cool the outlet gas from the water-cooled cooler.
Calculate the minimum amount of cooling energy that the high-temperature section must provide to the low-temperature section when the cooling demand required by the user is 4000 kW.

Options:
A. 3800kW
B. 5600kW
C. 4400kW
D. 5200kW
E. 3600kW
F. 6200kW
G. 5000kW
H. 6000kW
I. 4200kW
J. 4800kW
Answer: J
It is known that the pressure of the compressed air entering the unit is 607.8 kPa and the temperature is 300 K, while the environmental atmospheric pressure is 101.3 kPa and the temperature is 300 K. Assume that the air is composed only of two gases, nitrogen and oxygen, with a molar fraction of 20.9% oxygen and 79.1% nitrogen. The pressure and temperature of the separated product are the same as that of the atmospheric environment, with a purity of 98.5% nitrogen and 99.1% oxygen by molar fraction.
Calculate the exergy loss and exergy efficiency of a low-pressure air separation unit.

Options:
A. $$2800kJ/kmol$$
$$0.290$$
B. $$3184kJ/kmol$$
$$0.256$$
C. $$3000kJ/kmol$$
$$0.230$$
D. $$2900kJ/kmol$$
$$0.180$$
E. $$3200kJ/kmol$$
$$0.200$$
F. $$3100kJ/kmol$$
$$0.100$$
G. $$3500kJ/kmol$$
$$0.220$$
H. $$3300kJ/kmol$$
$$0.150$$
I. $$3400kJ/kmol$$
$$0.275$$
J. $$3250kJ/kmol$$
$$0.210$$
Answer: B
The resistance on a spherical bubble with radius $a$ moving slowly in a straight line at constant velocity in a homogeneous, incompressible fluid is ( ) .
( Assume body forces are negligible and the viscosity coefficient $\mu$ is constant. )

Options:
A. $$
5 \pi\mu a^2
$$
B. $$
\pi\mu a^3 U
$$
C. $$
4 \pi\mu U^2
$$
D. $$
4 \pi a U
$$
E. $$
2 \pi\mu a^2 U
$$
F. $$
8 \pi\mu a U
$$
G. $$
4 \pi\mu a U
$$
H. $$
12 \pi\mu a
$$
I. $$
6 \pi\mu a U
$$
J. $$
3 \pi\mu a U
$$
Answer: G
Consider the Rankine combined vortex in the plane steady motion of an ideal incompressible homogeneous fluid. If body forces are neglected, the velocity distribution of the fluid is
$$
v = \left\{\begin{array} {l} {\omega r \mathbf{\theta_{0}},\ r<r_{0},} \\ {\displaystyle\frac{\Gamma} {2 \pi r} \mathbf{\theta_{0}},\ r>r_{0},} \\ \end{array} \right.
$$
where $r_{0}$ is the core radius of the vortex, and $\mathbf{\theta_{0}}$ is the unit vector in the $\theta$ direction in plane polar coordinates. Assume that the velocity is continuous at $r = r_{0}$, and the fluid pressure at infinity is given. Here, $\omega$ and $\Gamma$ are constants.The pressure distribution within the vortex core is ( ) .

Options:
A. $$
p_{\infty}+\frac{1} {4} \rho\omega^{2} ( r^{2}-r_{0}^{2} )
$$
B. $$
p_{\infty}+\frac{1} {2} \rho\omega^{2} ( r^{2}-r_{0}^{2} )
$$
C. $$
p_{\infty}+\frac{1} {2} \rho\omega^{2} ( r_{0}^{2}+2 r^{2} )
$$
D. $$
p_{\infty}+\frac{1} {2} \rho\omega ( r^{2}-r_{0}^{2} )
$$
E. $$
p_{\infty}+\frac{\rho} {2} \omega^{2} ( r^{2}-2 r_{0}^{2} )
$$
F. $$
p_{\infty}+\frac{1} {2} \rho\omega^{2} ( r^{2}-2 r_{0}^{2} )
$$
G. $$
p_{\infty}+\frac{1} {5} \rho\Gamma^{2} ( r^{2}-2 r_{0}^{2} )
$$
H. $$
p_{\infty}+\frac{1} {2} \rho\omega^{2} ( r_{0}^{2}-2 r^{2} )
$$
I. $$
p_{\infty}+\frac{1} {2} \rho\omega^{3} ( r^{2}-2 r_{0}^{2} )
$$
J. $$
p_{\infty}+\frac{1} {3} \rho\omega^{2} ( r^{2}-2 r_{0}^{2} )
$$
Answer: F
A bubble with a radius of $5 \times 10^{-4} \mathrm{cm}$ is in the water, where the absolute pressure is 1.2 atmospheres. What is the pressure of the air inside the bubble ?
( Assume the surface tension coefficient of water is $\alpha = 72.5 \times 10^{-5} \mathrm{N/cm}$. )

Options:
A. $$
1. 8 0 3 2 \times1 0^{5} \mathrm{P a}
$$
B. $$
1. 7 0 2 4 \times1 0^{5} \mathrm{P a}
$$
C. $$
2. 1 9 6 5 \times1 0^{4} \mathrm{P a}
$$
D. $$
1. 3 7 3 4 \times1 0^{5} \mathrm{P a}
$$
E. $$
1. 9 5 6 2 \times1 0^{4} \mathrm{P a}
$$
F. $$
1. 2 1 4 7 \times1 0^{5} \mathrm{P a}
$$
G. $$
1. 5 0 9 2 \times1 0^{3} \mathrm{P a}
$$
H. $$
1. 4 0 8 7 \times1 0^{3} \mathrm{P a}
$$
I. $$
1. 5 0 5 9 \times1 0^{5} \mathrm{P a}
$$
J. $$
1. 6 4 6 8 \times1 0^{5} \mathrm{P a}
$$
Answer: I
There is a cylindrical buoy, 1.8m in height and 1.35m in diameter, with a mass of 770kg, and its center of gravity is 0.9m from the bottom. Prove that this buoy cannot stably float upright in seawater with a density of 1025kg/m³. If a chain is attached to the bottom of the cylinder, what is the minimum downward force needed on the chain to allow the buoy to float upright stably?

Options:
A. $$ 8 9 2 1 \mathrm{N}_{\bullet} $$
B. $$ 7 8 5 9 \mathrm{N}_{\bullet} $$
C. $$ 1 7 8 0 \mathrm{N}_{\bullet} $$
D. $$
4 6 3 6 \mathrm{N}
$$
E. $$ 6 5 1 2 \mathrm{N}_{\bullet} $$
F. $$ 7 1 4 6 \mathrm{N}_{\bullet} $$
G. $$ 5 4 8 3 \mathrm{N}_{\bullet} $$
H. $$ 3 4 2 7 \mathrm{N}_{\bullet} $$
I. $$ 2 5 4 1 \mathrm{N}_{\bullet} $$
J. $$ 9 3 6 8 \mathrm{N}_{\bullet} $$
Answer: D
Given the stream function for an incompressible homogeneous fluid in planar steady flow as
$$
\psi=x^{2}+y^{2}+y,
$$
find the difference in total mechanical energy $\left( \frac{p} {\rho}+\frac{V^{2}} {2} + g z \right)$ per unit mass of fluid between the points (1,1) and (2,2).

Options:
A. The difference in mechanical energy between the two points is $7$.
B. The difference in mechanical energy between the two points is $3$.
C. The difference in mechanical energy between the two points is $12$.
D. The difference in mechanical energy between the two points is $11$.
E. The difference in mechanical energy between the two points is $6$.
F. The difference in mechanical energy between the two points is $4$.
G. The difference in mechanical energy between the two points is $9$.
H. The difference in mechanical energy between the two points is $10$.
I. The difference in mechanical energy between the two points is $8$.
J. The difference in mechanical energy between the two points is $5$.
Answer: A
The flow around an ovoid can be simulated by a combination of the following basic flows: a uniform flow $U$ in the direction along the positive $x$ axis, a point source with strength $2\pi Q$ located at $z=-a$, and a point sink with strength $2\pi Q$ located at $z=a$. Find the location of the stagnation point and the equation of the ovoid.

Options:
A. $$
\pm\sqrt{a^{2}-\frac{2 a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{2 a x} {y^{2}-x^{2}-a^{2}}=0
$$
B. $$
\pm\sqrt{2 a^{2}+\frac{2 a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{2 a y} {y^{2}-x^{2}-a^{2}}=0
$$
C. $$
\pm\sqrt{a^{2}+\frac{4a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{2 a y} {y^{2}-x^{2}-a^{2}}=0
$$
D. $$
z = \pm\sqrt{a^{2}+\frac{2 a Q} {U}}
$$
$$
U y-Q \mathrm{a r c t a n} \frac{2 a y} {x^{2}-y^{2}-a^{2}}=0
$$
E. $$
\pm\sqrt{a^{3}-\frac{a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{a x} {y^{2}-x^{2}-a^{3}}=0
$$
F. $$
\pm\sqrt{a^{2}+\frac{3 a Q} {U}}
$$
$$
U y-Q \mathrm{a r c t a n} \frac{2 a y} {x^{2}-y^{2}-a^{2}}=0
$$
G. $$
\pm\sqrt{a^{3}+\frac{3 aQ} {U}}
$$
$$
U y-Q \mathrm{a r c t a n} \frac{2 a x} {x^{2}-y^{2}-a^{2}}=0
$$
H. $$
\pm\sqrt{a^{2}+\frac{a Q} {2U}}
$$
$$
U y-Q \mathrm{a r c t a n} \frac{4 a y} {x^{2}-a^{2}-y^{2}}=0
$$
I. $$
\pm\sqrt{a^{2}+\frac{4 a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{2 a x} {x^{2}-a^{2}-y^{2}}=0
$$
J. $$
\pm\sqrt{2 a^{2}+\frac{4 a Q} {U}}
$$
$$
U x-Q \mathrm{a r c t a n} \frac{2 a y} {x^{2}-y^{2}-a^{2}}=0
$$
Answer: D
In an ideal incompressible homogeneous fluid, there is a vortex tube with a radius of $a$, whose axis is along the vertical $z$-axis. Within this vortex tube, the fluid motion is the same as that of a rigid body, i.e., the vorticity is constant. If the fluid has a free surface and the fluid is stationary at infinity, determine the shape of the free surface.

Options:
A. When $r > a$, $z = -\frac{a^{4} \omega^{2}}{g r^{3}}$
When $r < a$, $z = -\frac{a^{2} \omega^{3}}{g} \Big( 1 - \frac{r^{2}}{2 a^{2}} \Big)$
B. When $r > a$, $z = -\frac{a^{4} \omega^{2}}{2 g r^{2}}$
When $r < a$, $z = -\frac{a^{2} \omega^{2}}{g} \Big( 1 - \frac{r^{2}}{2 a^{2}} \Big)$
C. When $r > a$, $z = -\frac{a^{4} \omega^{2}}{g r^{3}}$
When $r < a$, $z = -\frac{a^{2} \omega^{2}}{g} \Big( 1 - \frac{r}{2 a} \Big)$
D. When $r > a$, $z = -\frac{a^{4} \omega}{g r^{2}}$
When $r < a$, $z = -\frac{a^{2} \omega}{g} \Big( 1 - \frac{r}{2 a^{2}} \Big)$
E. When $r > a$, $z = -\frac{\omega a^{4}}{2 g r^{2}}$
When $r < a$, $z = -\frac{\omega a^{2}}{g} \Big( 1 - \frac{r^{2}}{2 a^{2}} \Big)$
F. When $r > a$, $z = -\frac{a^{4} \omega^{2}}{2 g r}$
When $r < a$, $z = -\frac{a^{2} \omega^{2}}{g} \Big( \frac{r^{2}}{2 a^{2}} - 1 \Big)$
G. When $r > a$, $z = -\frac{a^{4} \omega}{2 g r^{2}}$
When $r < a$, $z = -\frac{a^{2} \omega}{g} \Big( 1 - \frac{r^{2}}{2 a^{2}} \Big)$
H. When $r > a$, $z = -\frac{a^{4} \omega^{2}}{g r^{2}}$
When $r < a$, $z = -\frac{a^{2} \omega^{2}}{g} \Big( 1 + \frac{r^{2}}{2 a^{2}} \Big)$
I. When $r > a$, $z = -\frac{a^{5} \omega^{2}}{2 g r^{2}}$
When $r < a$, $z = -\frac{a^{3} \omega^{2}}{g} \Big( 1 - \frac{r^{2}}{2 a^{3}} \Big)$
J. When $r > a$, $z = -\frac{a^{3} \omega^{2}}{2 g r^{2}}$
When $r < a$, $z = -\frac{a^{2} \omega^{2}}{g} \Big( 1 - \frac{r^{3}}{a^{3}} \Big)$
Answer: B
The mean activity of a 0.1 mol/kg $Na_{\mathrm{2}}$$SO_{\mathrm{4}}$ solution is().

Options:
A. $$
= 0. 1 0 5
$$
B. $$
= 0. 0 6 8
$$
C. $$
= 0. 0 4 5
$$
D. $$
= 0. 0 9 6
$$
E. $$
= 0. 1 5 3
$$
F. $$
= 0. 0 3 4
$$
G. $$
= 0. 0 7 2
$$
H. $$
= 0. 1 2 3
$$
I. $$
= 0. 0 8 0
$$
J. $$
= 0. 0 5 9
$$
Answer: G
$\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ represent 0.544 3 mol/dm$^{3}$ and 0.271 1 mol/dm$^{3}$ $H_{\mathrm{2}}$$SO_{\mathrm{4}}$, respectively. These are solutions in anhydrous methanol. At 25°C, the electromotive force of the cell Pt, $H_{\mathrm{2}}$(101,325 Pa) $\mathrm{S}_{1}$, $Hg_{\mathrm{2}}$$SO_{\mathrm{4}}$(solid) | Hg is 598.9 mV; the electromotive force of the cell Pt, $H_{\mathrm{2}}$(101,325 Pa) | $\mathrm{S}_{2}$, $Hg_{\mathrm{2}}$$SO_{\mathrm{4}}$(solid) | Hg is 622.6 mV; the electromotive force of the cell Hg | $Hg_{\mathrm{2}}$$SO_{\mathrm{4}}$(solid), $S_{\mathrm{1}}$ | $S_{\mathrm{2}}$, $Hg_{\mathrm{2}}$$SO_{\mathrm{4}}$(solid) | Hg is 17.49 mV.
Try to find the transference numbers of the sulfate anion and cation in anhydrous methanol solution at 25°C. Assume that the transference numbers are independent of concentration and that sulfur exists only in the form of the sulfate ion in this solution is().

Options:
A. $$
t_{+}=0. 8 1
$$
$$
t_{-}=0. 1 9
$$
B. $$
t_{+}=0. 5 2
$$
$$
t_{-}=0. 4 8
$$
C. $$
t_{+}=0. 2 8
$$
$$
t_{-}=0. 7 2
$$
D. $$
t_{+}=0. 5 7
$$
$$
t_{-}=0. 4 3
$$
E. $$
t_{+}=0. 4 5
$$
$$
t_{-}=0. 5 5
$$
F. $$
t_{+}=0. 7 4
$$
$$
t_{-}=0. 2 6
$$
G. $$
t_{+}=0. 3 9
$$
$$
t_{-}=0. 6 1
$$
H. $$
t_{+}=0. 6 6
$$
$$
t_{-}=0. 3 4
$$
I. $$
t_{+}=0. 8 8
$$
$$
t_{-}=0. 1 2
$$
J. $$
t_{+}=0. 6 3
$$
$$
t_{-}=0. 3 7
$$
Answer: F
There are two cells:
(a) Ag|AgCl(solid), $S_{\mathrm{1}}$|$H_{\mathrm{2}}$, Pt--Pt, H,|$S_{\mathrm{2}}$, AgCl| (solid) Ag
(b) Ag|AgCl(solid), $S_{\mathrm{1}} | S_{\mathrm{2}}$, AgCl(solid) Ag
It is known that $S_{1}$ is an ethanol solution of HCl with concentration 0.082 mol/kg, and $S_{2}$ is an ethanol solution of HCl with concentration 0.0082 mol/kg. Given the electromotive force $| E_{a} |= 82.2 \mathrm{mV}$ and the electromotive force $\left| E_{b} \right| = 57.9 \, \, \, \mathrm{mV}_{\circ}$ at 25°C, the transference number of $\mathrm{H}^{+}$ ions in the dilute HCl ethanol solution is().

Options:
A. $$
= 0. 6 8 3 1
$$
B. $$
= 0. 6 9 0 6
$$
C. $$
= 0. 6 4 7 3
$$
D. $$
= 0. 8 1 5 2
$$
E. $$
= 0. 5 9 2 8
$$
F. $$
= 0. 7 9 1 0
$$
G. $$
= 0. 7 0 4 4
$$
H. $$
= 0. 5 7 3 9
$$
I. $$
= 0. 7 3 8 7
$$
J. $$
= 0. 6 2 4 5
$$
Answer: G
For the cell $(-) \, \mathrm{Zn} \, | \, \mathrm{ZnCl}_{2} (1 \, \mathrm{mol/kg}) \, || \, \mathrm{HCl} (a=1) \, | \, \mathrm{H}_{2} (\mathrm{101,325} \, \mathrm{Pa}) \, \mathrm{Pt} \, (+)$, when a current of $10^{-4} \, \mathrm{A/cm}^{2}$ flows from the positive electrode to the negative electrode through the external circuit, the voltage across the cell is 1.24 $\mathrm{V}$. If the ohmic drop in the solution is 0.1 $\mathrm{V}$ and the average activity coefficient in the 1 mol/L ZnCl$_{2}$ aqueous solution is $\gamma_{\pm} = 0.33$, determine the overpotential of the zinc electrode at this current density, given that the overpotential at the hydrogen electrode is 0.164 $\mathrm{V}$ for $j = 10^{-4} \, \mathrm{A/cm}^{2}$ is().

Options:
A. $$
= 0. 2 0 8 V
$$
B. $$
= 0. 1 5 9 V
$$
C. $$
= 0. 1 7 9 V
$$
D. $$
= 0. 1 8 7 V
$$
E. $$
= 0. 2 0 4 V
$$
F. $$
= 0. 1 9 9 V
$$
G. $$
= 0. 1 6 3 V
$$
H. $$
= 0. 1 9 2 V
$$
I. $$
= 0. 2 0 1 V
$$
J. $$
= 0. 2 1 0 V
$$
Answer: F
Given that at 25°C, the cathodic reaction $0 + 2 \mathrm{e} \to \mathrm{R}$ is controlled by the diffusion step, both $\mathrm{O}$ and $\mathbf{R}$ are soluble, with $c_{\mathrm{O}}^{0}=0.1 \mathrm{~mol/L}$ and $c_{\mathrm{R}}^{0}=0$. The thickness of the diffusion layer is 0.01 cm, and the diffusion coefficient of $\mathrm{O}$ is $1.5 \times 10^{-4} \mathrm{~cm^{2}/s}$. What is the cathode potential when $j_{\mathrm{c}} = 0.2 \mathrm{~A/cm^{2}}$?

Options:
A. $$
=-0. 2 0 1 V
$$
B. $$
=-0. 1 4 2 V
$$
C. $$
=-0. 1 3 9 V
$$
D. $$
=-0. 1 8 3 V
$$
E. $$
=-0. 1 1 9 V
$$
F. $$
=-0. 1 6 0 V
$$
G. $$
=-0. 1 5 8 V
$$
H. $$
=-0. 0 9 5 V
$$
I. $$
=-0. 1 4 5 V
$$
J. $$
=-0. 1 2 7 V
$$
Answer: B
Install a zinc plate as a sacrificial anode on a steel hull; during corrosion in seawater, zinc dissolves. At 25°C, the exchange current density for the reaction $\mathrm{Zn^{2+} + 2e^-} \rightleftharpoons \mathrm{Zn}$ is $2 \times 10^{-5} \mathrm{A/cm}^2$, and the transfer coefficient $\alpha = 0.9$. The rate of zinc anode dissolution and the polarization resistance value at 25°C when the anode polarization is 0.05V is().

Options:
A. $$
4 8 0. 2 \Omega/ c m^{2}
$$
B. $$
3 6 8. 7 \Omega/ c m^{2}
$$
C. $$
3 9 0. 4 \Omega/ c m^{2}
$$
D. $$
5 0 2. 2 \Omega/ c m^{2}
$$
E. $$
5 7 3. 3 \Omega/ c m^{2}
$$
F. $$
4 4 2. 5 \Omega/ c m^{2}
$$
G. $$
4 1 0. 6 \Omega/ c m^{2}
$$
H. $$
4 9 5. 1 \Omega/ c m^{2}
$$
I. $$
5 2 1. 9 \Omega/ c m^{2}
$$
J. $$
3 2 5. 8 \Omega/ c m^{2}
$$
Answer: F
A linearly polarized plane wave is incident from free space onto a medium interface with $\varepsilon_{r}=4 \,, \mu_{r}=1$. If the angle between the electric field of the incident wave and the plane of incidence is 45°, What is the value of the incident angle $θ_{\mathrm{i}}$ when the reflected wave has only a perpendicular polarization?

Options:
A. $$\left( {\sqrt{2}\, \mathrm{E}_{i 0}} \right)
$$
B. $$\left( {\mathrm{E}_{i 0}} \right)
$$
C. $$\left( {\frac{\sqrt{3} \mathrm{E}_{i 0}} {2}} \right)
$$
D. $$\left( {\frac{\mathrm{E}_{i 0}} {\sqrt{2}}} \right)
$$
E. $$\left( {2\,\mathrm{E}_{i 0}} \right)
$$
F. $$\left( {\frac{\mathrm{E}_{i 0}} {\sqrt{5}}} \right)
$$
G. $$\left( {\frac{\mathrm{E}_{i 0}} {2}} \right)
$$
H. $$\left( {\sqrt{3}\, \mathrm{E}_{i 0}} \right)
$$
I. $$\left( {\frac{\mathrm{E}_{i 0}} {\sqrt{3}}} \right)
$$
J. $$\left( {\frac{\mathrm{E}_{i 0}} {\sqrt{4}}} \right)
$$
Answer: D
Consider a model of the universe consisting of neutral hydrogen atoms for which the average (rms) speed of the atoms is 600 km $\mathrm{s}^{-1}$ (approximately the speed of the Local Group relative to the Hubble Flow). Show that $\rho\gg P / c^{2}$ for the gas. For an adiabatically expanding universe, for what value of $R$ and $z$ will $\rho=P / c^{2}$ ?

Options:
A. $$
1.2 \times1 0^{1 3}.
$$
B. $$
9 \times1 0^{1 2}.
$$
C. $$
7 \times1 0^{1 1}.
$$
D. $$
4 \times1 0^{1 2}.
$$
E. $$
2.5 \times1 0^{1 3}.
$$
F. $$
3.5 \times1 0^{1 3}.
$$
G. $$
5 \times1 0^{1 2}.
$$
H. $$
8 \times1 0^{1 2}.
$$
I. $$
1.5 \times1 0^{1 2}.
$$
J. $$
6 \times1 0^{1 1}.
$$
Answer: D
The color change of green tea soup is primarily due to the oxidation of polyphenols, flavonoids, and their ().

Options:
A. Proteins
B. Chlorophylls
C. Carotenoids
D. Alkaloids
E. Terpenoids
F. Lignans
G. Anthocyanins
H. Glycosides
I. Amino acids
J. Tannins
Answer: H
A homogeneous disc with a radius of $r$ rotates about its center of mass, starting with an angular velocity of $\omega_{0}$, and the axis forms an angle $\alpha$ with the normal to the disc surface. The angle between the precession axis and the normal to the disc surface is ().

Options:
A. $$
\omega_{0} \operatorname{c o s} \alpha{\sqrt{3+\operatorname{t a n}^{2} \alpha}}=\omega_{0} {\sqrt{1+4 \operatorname{c o s}^{2} \alpha}}
$$
B. $$
\omega_{0} \operatorname{c o t} \alpha{\sqrt{4+\operatorname{t a n}^{2} \alpha}}=\omega_{0} {\sqrt{1-3 \operatorname{c o t}^{2} \alpha}}
$$
C. $$
\omega_{0} \operatorname{t a n} \alpha{\sqrt{4+\operatorname{c o s}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{s i n}^{2} \alpha}}
$$
D. $$
\omega_{0} \operatorname{c o s} \alpha{\sqrt{1+\operatorname{t a n}^{2} \alpha}}=\omega_{0} {\sqrt{4+3 \operatorname{c o s}^{2} \alpha}}
$$
E. $$
\omega_{0} \operatorname{s i n} \alpha{\sqrt{4+\operatorname{c o s}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{s e c}^{2} \alpha}}
$$
F. $$
\omega_{0} \operatorname{c o s} \alpha{\sqrt{4+\operatorname{c o t}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{c o t}^{2} \alpha}}
$$
G. $$
\omega_{0} \operatorname{s e c} \alpha{\sqrt{4+\operatorname{s i n}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{s e c}^{2} \alpha}}
$$
H. $$
\omega_{0} \operatorname{t a n} \alpha{\sqrt{4+\operatorname{c o s}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{t a n}^{2} \alpha}}
$$
I. $$
\omega_{0} \operatorname{s i n} \alpha{\sqrt{4+\operatorname{s e c}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{s i n}^{2} \alpha}}
$$
J. $$
\omega_{0} \operatorname{c o s} \alpha{\sqrt{4+\operatorname{t a n}^{2} \alpha}}=\omega_{0} {\sqrt{1+3 \operatorname{c o s}^{2} \alpha}}
$$
Answer: J
The radius of a hemispherical bowl is $R$, and a rod with length $l$ has its end $A$ sliding inside the bowl while leaning against the edge at point $P$. If the rod always remains within a fixed vertical plane, what are the paths of any point on the rod and the instantaneous center of rotation?

Options:
A. $$
r=R \operatorname{c o s} \theta-2a
$$
$$
x^{2}+y^{2}=2 R^{2} \,
$$
B. $$
r=R \operatorname{c o s} \theta+a
$$
$$
x^{2}+y^{2}=6 R^{2} \,
$$
C. $$
r=R \operatorname{c o s} \theta-b
$$
$$
x^{2}+y^{2}=8 R^{2} \,
$$
D. $$
r=5 R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=3 R^{2} \,
$$
E. $$
r=R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=R^{2} \,
$$
F. $$
r=2 R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=4 R^{2} \,
$$
G. $$
r=4 R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=5 R^{2} \,
$$
H. $$
r=R \operatorname{c o s} \theta+2a
$$
$$
x^{2}+y^{2}=3 R^{2} \,
$$
I. $$
r=3 R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=R^{2} \,
$$
J. $$
r=7 R \operatorname{c o s} \theta-a
$$
$$
x^{2}+y^{2}=R^{2} \,
$$
Answer: F
When a stationary electron-positron pair (i.e., an electron and a positron) annihilates, it produces two photons. What is the maximum velocity that one of these photons can impart to another stationary electron upon collision?

Options:
A. $$
v=0. 6 c
$$
B. $$
v=0. 4 c
$$
C. $$
v=0. 7 c
$$
D. $$
v=0. 5 c
$$
E. $$
v=0. 8 c
$$
F. $$
v=1. 1 c
$$
G. $$
v=0. 9 c
$$
H. $$
v=0. 3 c
$$
I. $$
v=1. 0 c
$$
J. $$
v=0. 2 c
$$
Answer: E
A heavy tuning fork approaches the wall at a speed of $u=25 \mathrm{cm/s}$, with the tuning fork situated between a stationary observer and the wall. The observer hears a beat frequency of $\nu=3 \mathrm{s}^{-1}$. If the speed of sound is $v=340 \mathrm{m/s}$, what is the vibration frequency of the tuning fork?

Options:
A. $$
\approx2 0 0 0 \mathrm{H z} \;
$$
B. $$
\approx2 0 3 0 \mathrm{H z} \;
$$
C. $$
\approx2 0 7 5 \mathrm{H z} \;
$$
D. $$
\approx2 1 5 0 \mathrm{H z} \;
$$
E. $$
\approx2 2 0 0 \mathrm{H z} \;
$$
F. $$
\approx2 0 9 0 \mathrm{H z} \;
$$
G. $$
\approx2 0 6 5 \mathrm{H z} \;
$$
H. $$
\approx2 1 0 0 \mathrm{H z} \;
$$
I. $$
\approx2 0 4 0 \mathrm{H z} \;
$$
J. $$
\approx1 9 8 0 \mathrm{H z} \;
$$
Answer: I
A uniform cylinder with a diameter of 1.0 cm rolls on flat ground and stops after 5.0 s, covering a distance of 15 m before stopping. What is the rolling friction coefficient $k$? $k$ is defined as the ratio of the rolling friction torque $L$ at the point of contact to the normal force $N$, $k = L / N$.

Options:
A. $$ 0.085cm $$
B. $$ 0.095cm $$
C. $$ 0.088cm $$
D. $$ 0.094cm $$
E. $$ 0.098cm $$
F. $$ 0.096cm $$
G. $$
0.092cm
$$
H. $$ 0.099cm $$
I. $$ 0.080cm $$
J. $$ 0.090cm $$
Answer: G
A separately excited DC motor has nameplate data of $\mathrm{P}_{\mathrm{N}}=96 \mathrm{kW}, \mathrm{U}_{\mathrm{N}}=440 \mathrm{V}, \mathrm{I}_{\mathrm{N}}=255 \mathrm{A}, \mathrm{I}_{\mathrm{N}}=5 \mathrm{A}$ $\mathrm{n}_{\mathrm{N}}=1550 \, \mathrm{r} \setminus\mathrm{min}$, and it is known that $\mathrm{R_{a}=0.078 \, \Omega}$. What is the rated electromagnetic torque $\mathrm{T_{em}}$ of the motor?

Options:
A. 678.9N·m
B. 649.5N·m
C. 657.4N·m
D. 540.3N·m
E. 612.8N·m
F. 644.6N·m
G. 631.7N·m
H. 695.2N·m
I. 662.5N·m
J. 623.1N·m
Answer: F
Two single-phase transformers, with a voltage of 220V/110V, have equal turns on the primary side, but the no-load current $\mathbf{I}_{\mathrm{O I}} = 2 \mathbf{I}_{\mathrm{O I I}}$. Now, if the primary windings of the two transformers are connected in series with the same polarity and a voltage of 440V is applied, what will be the no-load voltage on the secondary side of each transformer?

Options:
A. 330.0V,165.0V
B. 180.0V,90.0V
C. 146.7V,73.4V
D. 190.0V,95.0V
E. 250.0V,125.0V
F. 300.0V,150.0V
G. 240.0V,120.0V
H. 120.0V,60.0V
I. 200.0V,100.0V
J. 220.0V,110.0V
Answer: C
The velocity vector of a certain flow field can be expressed as
$$
u ( x, y, z, \theta)=x y z i+y j-3 z \theta k
$$
Try to find the acceleration vector at the point (2, 1, 2, 1).

Options:
A. $$
= 3 x+2 k i
$$
B. $$
= x y+z \theta i
$$
C. $$
= 4 y+3 z i
$$
D. $$
= x^2 i+y \theta j
$$
E. $$
= 2 i+3 j+1 k
$$
F. $$
= 3 i-x z j
$$
G. $$
= 1 j+1 2 k
$$
H. $$
= 2 y z j
$$
I. $$
= 0 i+2 y k
$$
J. $$
= 1 j-3 \theta z
$$
Answer: G
The formulas for calculating the local convective heat transfer coefficient $h_{x}$ for the laminar boundary layer and the turbulent boundary layer on a flat plate surface are respectively
$$
\begin{array} {c} {{{h_{x}=0.332 \frac{k} {x} R e_{x}^{1 / 2} P r^{1 / 3}}}} \\ {{{h_{x}=0.0292 \frac{k} {x} R e_{x}^{0.8} P r^{1 / 3}}}} \\ \end{array}
$$
Try to derive the expression for the average convective heat transfer coefficient $h_{\mathrm{m}}$ from the leading edge of the flat plate to the location $x=L$ within the turbulent boundary layer.

Options:
A. $$
R e_{x_{\mathrm{c}}}^{0. 8}-1 5. 2 5 R e_{x_{\mathrm{c}}}^{2 / 5}
$$
B. $$
R e_{x_{\mathrm{c}}}^{0. 8}-1 8. 1 9 R e_{x_{\mathrm{c}}}^{1 / 2}
$$
C. $$
R e_{x_{\mathrm{c}}}^{0. 7}-1 9. 0 5 R e_{x_{\mathrm{c}}}^{3 / 2}
$$
D. $$
R e_{x_{\mathrm{c}}}^{0. 4}-1 9. 4 9 R e_{x_{\mathrm{c}}}^{8 / 3}
$$
E. $$
R e_{x_{\mathrm{c}}}^{1. 1}-2 1. 8 3 R e_{x_{\mathrm{c}}}^{9 / 7}
$$
F. $$
R e_{x_{\mathrm{c}}}^{0. 6}-1 8. 5 5 R e_{x_{\mathrm{c}}}^{2 / 3}
$$
G. $$
R e_{x_{\mathrm{c}}}^{0. 5}-1 7. 0 0 R e_{x_{\mathrm{c}}}^{1 / 4}
$$
H. $$
R e_{x_{\mathrm{c}}}^{1. 0}-2 0. 1 1 R e_{x_{\mathrm{c}}}^{1 / 6}
$$
I. $$
R e_{x_{\mathrm{c}}}^{0. 9}-1 6. 8 2 R e_{x_{\mathrm{c}}}^{1 / 5}
$$
J. $$
R e_{x_{\mathrm{c}}}^{0. 2}-1 7. 9 5 R e_{x_{\mathrm{c}}}^{5 / 4}
$$
Answer: B
At $25^{\circ} \ \mathrm{C}$, under pressure between 0 and 1000 atm, the volume of water is measured as:
$$
V = 18.066 - 0.715 \times 10^{-3} \, \mathbf{p} + 0.046 \times 10^{-8} \, \mathbf{p}^{2} \, \mathrm{cm}^{3} \cdot \mathrm{mol}^{-1}
$$
Assuming the temperature remains constant, the work done by the surroundings when compressing 1 mol of water from 1 atm to 1000 atm is___.

Options:
A. $$
1 9. 8 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
B. $$
4 7. 2 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
C. $$
1 2. 5 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
D. $$
2 8. 6 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
E. $$
5 8. 4 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
F. $$
3 3. 1 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
G. $$
5 2. 0 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
H. $$
3 9. 3 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
I. $$
6 4. 7 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
J. $$
4 2. 9 \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
Answer: F
There is an electric dipole with an amplitude $p$ and frequency $\omega$ located at a distance $a / 2$ from an ideal conductor plane, with $p$ parallel to the conductor plane. Assuming $a \ll \lambda$, determine the electromagnetic field and radiation energy flow at a distance $R \gg \lambda$.

Options:
A. $$
\frac{\mu_{0} \omega^{6} a^{2} p_{0}^{2} \operatorname{s i n}^{2} \theta} {3 3 \pi^{2} c^{3} R^{2}} \left( \operatorname{c o s}^{2} \phi+\operatorname{s i n}^{2} \phi\operatorname{c o s}^{2} \theta\right)
$$
B. $$
\frac{\epsilon_{0} \omega^{6} a^{2} p_{0} \operatorname{s i n}^{2} \theta} {3 2 \pi^{2} c^{3} R^{2}} \left( \operatorname{c o s}^{2} \phi+\operatorname{s i n}^{2} \theta\operatorname{c o s}^{2} \phi\right)
$$
C. $$
\frac{\mu_{0} \omega^{4} p_{0} a^{3} \operatorname{s i n}^{2} \phi} {4 2 \pi^{2} c^{3} R^{2}} \left( \operatorname{c o s}^{2} \phi+\operatorname{c o s}^{2} \theta\operatorname{s i n}^{2} \phi\right)
$$
D. $$
\frac{\mu_{0} \omega^{6} a^{2} p_{0}^{2} \operatorname{s i n}^{2} \phi} {3 3 \pi^{2} c^{3} R^{2}} \left( \operatorname{c o s}^{2} \theta+\operatorname{s i n}^{2} \phi\operatorname{c o s}^{2} \theta\right)
$$
E. $$
\frac{\epsilon_{0} \omega^{5} a^{3} p_{0} \operatorname{c o s}^{2} \phi} {3 2 \pi^{3} c^{3} R^{2}} \left( \operatorname{s i n}^{2} \theta+\operatorname{s i n}^{2} \phi\operatorname{c o s}^{2} \theta\right)
$$
F. $$
\frac{\mu_{0} \omega^{4} a^{2} p_{0} \operatorname{s i n}^{2} \phi} {3 2 \pi^{2} c^{3} R^{3}} \left( \operatorname{c o s}^{2} \theta+\operatorname{c o s}^{2} \theta\operatorname{s i n}^{2} \phi\right)
$$
G. $$
\frac{\epsilon_{0} \omega^{5} a^{2} p_{0} \operatorname{s i n}^{2} \theta} {3 2 \pi^{3} c^{2} R^{3}} \left( \operatorname{c o s}^{2} \phi+\operatorname{s i n}^{2} \theta\operatorname{c o s}^{2} \phi\right)
$$
H. $$
\frac{\mu_{0} \omega^{6} a p_{0}^{2} \operatorname{c o s}^2 \phi} {3 2 \pi^{2} c^{2} R^{2}} \left( \operatorname{s i n}^{2} \theta+\operatorname{c o s}^{2} \theta\operatorname{s i n}^{2} \phi\right)
$$
I. $$
\frac{\mu_{0} \omega^{6} a^{2} p_{0}^{2} \operatorname{c o s}^{2} \theta} {3 2 \pi^{2} c^{3} R^{2}} \left( \operatorname{s i n}^{2} \phi+\operatorname{c o s}^{2} \theta\operatorname{c o s}^{2} \phi\right)
$$
J. $$
\frac{\epsilon_{0} \omega^{5} a^{2} p_{0} \operatorname{c o s}^2 \theta} {2 2 \pi^{2} c^{3} R^{2}} \left( \operatorname{s i n}^{2} \theta+\operatorname{s i n}^{2} \phi\operatorname{c o s}^{2} \theta\right)
$$
Answer: I
Which aspect of physical fitness characteristics determines the endurance of athletes in showcasing the competitive ability of aesthetic and challenging event group training?

Options:
A. Flexibility Training
B. Skill Technique
C. Anaerobic Capacity
D. High body fat
E. Physical Fitness
F. Extreme cold training
G. Speed Training
H. Bone Density
I. Muscle Strength
J. Height
Answer: E
Estimate the amount of mass lost by Comet Halley during its most recent trip through the inner Solar System. Take into consideration the fact that the comet exhibits significant activity only during a short period of time near perihelion (an interval of approximately one year). Assuming that the mass loss rates are the same for each trip, how many more trips might the comet be able to make before it becomes extinct?

Options:
A. $$
1 5 0 \mathrm{~ t r i p s}.
$$
B. $$
1 0 0 \mathrm{~ t r i p s}.
$$
C. $$
2 4 0 \mathrm{~ t r i p s}.
$$
D. $$
6 0 0 \mathrm{~ t r i p s}.
$$
E. $$
1 2 5 \mathrm{~ t r i p s}.
$$
F. $$
1 8 0 \mathrm{~ t r i p s}.
$$
G. $$
9 0 ~ \mathrm{t r i p s}.
$$
H. $$
9 5 0 \mathrm{~ t r i p s}.
$$
I. $$
7 5 0 \mathrm{~ t r i p s}.
$$
J. $$
3 5 0 \mathrm{~ t r i p s}.
$$
Answer: E
An infinitely long uniformly charged cylindrical body with a radius of $a$ has a charge volume density of $\rho$. Within this cylinder, an infinitely long parallel cylindrical cavity with a radius of $a \, b$ is carved out. The distance between the axes of the two cylinders is $d$. The electric field intensity at all points inside the cavity is ().

Options:
A. $$
\frac{\rho d}{2 \pi \varepsilon_{0}}
$$
B. $$
\frac{\rho d r_{a b}^{°}} {2 \varepsilon_{0}}
$$
C. $$
\frac{\rho a d}{\varepsilon_{0}}
$$
D. $$
\frac{\rho d r_{a b}}{2 \pi \varepsilon_{0}}
$$
E. $$
\frac{\rho d^{2} r_{a b}}{2 \varepsilon_{0}}
$$
F. $$
\frac{\rho d}{\varepsilon_{0} r_{a b}}
$$
G. $$
\frac{2 \rho d r_{a b}}{\varepsilon_{0}}
$$
H. $$
\frac{\rho}{2 \varepsilon_{0} r_{a b}^{°}}
$$
I. $$
\frac{\rho r_{a} b}{2 \varepsilon_{0}}
$$
J. $$
\frac{\rho d a b}{4 \varepsilon_{0} \pi}
$$
Answer: B
Let the reference point of the vector potential be at infinity. The vector potential of a straight-line current $I$ of length 2 meters at a point 1 meter away from the line current on its perpendicular line is ().

Options:
A. $$
e_{2} \, {\frac{\mu_{0} I} {4 \pi}} \operatorname{l n} {\frac{1+\sqrt{2}} {1-\sqrt{2}}}
$$
B. $$
e_{2} \, {\frac{\mu_{0} I} {4 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
C. $$
e_{1} \, {\frac{\mu_{0} I} {4 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
D. $$
e_{2} \, {\frac{\mu_{0} I} {2 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
E. $$
e_{2} \, {\frac{\mu_{0} I} {4 \pi}} \operatorname{l n} {\frac{\sqrt{2}-1} {\sqrt{2}+1}}
$$
F. $$
e_{1} \, {\frac{\mu_{0} I} {8 \pi}} \operatorname{l n} {\frac{\sqrt{2}-1} {\sqrt{2}+1}}
$$
G. $$
e_{2} \, {\frac{\mu_{0} I} {4 \pi}} \operatorname{l n} {\frac{\sqrt{3}+1} {\sqrt{3}-1}}
$$
H. $$
e_{2} \, {\frac{\mu_{0} I} {8 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
I. $$
e_{1} \, {\frac{\mu_{0}} {4 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
J. $$
e_{1} \, {\frac{\mu_{0} I} {2 \pi}} \operatorname{l n} {\frac{\sqrt{2}+1} {\sqrt{2}-1}}
$$
Answer: B
Let the distance between the plates of the parallel plate capacitor be $d$, the permittivity of the dielectric be $\mathbf{ε}_{0}$, and an AC power supply is connected between the plates with a voltage of $u = U_{\mathrm{m}} \sin \omega t$. What is the displacement current density at any point between the plates?

Options:
A. $$J_{D}={\frac{\partial E} {\partial t}}={\frac{2\varepsilon_{0} U_{m}} {d}} \omega\operatorname{c o s} \omega t
$$
B. $$J_{D}={\frac{\partial D} {\partial t}}={\frac{\varepsilon_{0} U_{m}} {2d}} \omega\operatorname{s i n} \omega t
$$
C. $$J_{D}={\frac{\partial E} {\partial t}}={\frac{\varepsilon_{0} (U_{m})} {d^{2}}} \omega\operatorname{c o s} \omega t
$$
D. $$J_{D}={\frac{\partial D} {\partial e}}={\frac{\varepsilon_{0} U_{m}} {2d}} \omega\operatorname{c o s} \omega t
$$
E. $$J_{D}={\frac{\partial D} {\partial t}}={\frac{\varepsilon_{0} U_{m}} {d^{3}}} \omega\operatorname{c o s} \omega t
$$
F. $$J_{D}={\frac{\partial D} {\partial e}}={\frac{\varepsilon_{0} U_{m}} {d}} \omega\operatorname{c o s} \omega t
$$
G. $$J_{D}={\frac{\partial D} {\partial e}}={\frac{\varepsilon_{0} U_{m}} {d}} \omega\operatorname{s i n} (\omega + t)
$$
H. $$J_{D}={\frac{\partial E} {\partial t}}={\frac{\varepsilon_{0} U_{m}} {d}} \frac{\omega^{2}} {c o s} \omega t
$$
I. $$J_{D}={\frac{\partial E} {\partial t}}={\frac{\varepsilon_{0} U_{m}^{2}} {d}} \omega\operatorname{c o s} \omega t
$$
J. $$J_{D}={\frac{\partial D} {\partial t}}={\frac{\varepsilon_{0} U_{m}} {d}} \omega\operatorname{s i n} \omega t
$$
Answer: F
A capacitor is composed of two concentric spherical plates with inner and outer radii of $a$ and $b$, respectively. The dielectric constant of the material between the plates is $\varepsilon_{0}$. When the charges on the inner and outer electrodes are ±$q$, what is the electrostatic energy stored in the capacitor?

Options:
A. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}} \biggl( \frac{1} {b}-\frac{1} {a} \biggr)
$$
B. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}} \biggl( \frac{1} {a}-\frac{1} {b} \biggr)
$$
C. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}} \biggl( \frac{b-a} {ab} \biggr)
$$
D. $$
\frac{q^{2}} {16 \pi\varepsilon_{0}} \biggl( \frac{1} {a^{2}}-\frac{1} {b^{2}} \biggr)
$$
E. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}ab} \biggl( b+a \biggr)
$$
F. $$
\frac{q^{2}} {16 \pi\varepsilon_{0}} \biggl( \frac{a} {b}+\frac{b} {a} \biggr)
$$
G. $$
\frac{q^{2}} {4 \pi\varepsilon_{0}} \biggl( b-a \biggr)
$$
H. $$
\frac{q^{2}} {4 \pi\varepsilon_{0}} \biggl( \frac{1} {a}+\frac{1} {b} \biggr)
$$
I. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}} \biggl( \ln{\frac{b}{a}} \biggr)
$$
J. $$
\frac{q^{2}} {8 \pi\varepsilon_{0}} \biggl( ab \biggr)
$$
Answer: B
In industrial production, $\mathrm{C_6 H_6}$ undergoes catalytic oxidation on a catalyst with $\mathrm{V_2 O_5}$ as the active component to produce $\mathrm{C_4 H_2 O_3}$ (maleic anhydride). To carry out development research, experiments were conducted in a single-tube fixed-bed reactor. The hourly air feed rate was 2,000 L (S.T.P), and the amount of $\mathrm{C_6 H_6}$ entering was $79 \, \mathrm{mL/h}$. At the reactor outlet, the gas composition was as follows: $\mathrm{C_6 H_6}$ concentration of $3 \times 10^{-4}$ (volume fraction), CO and $\mathrm{CO_2}$ concentration of 0.015 (volume fraction), $\mathrm{C_4 H_2 O_3}$ concentration of 0.0109 (volume fraction), with the rest being air. What is the concentration of $\mathrm{C_6 H_6}$ in the feed gas mixture? (Expressed as mole fraction).

Options:
A. $7.80 \times 10^{-3}$ (volume fraction)
B. $1.00 \times 10^{-2}$ (volume fraction)
C. $1.15 \times 10^{-2}$ (volume fraction)
D. $7.15 \times 10^{-3}$ (volume fraction)
E. $9.98 \times 10^{-3}$ (volume fraction)
F. $8.50 \times 10^{-3}$ (volume fraction)
G. $1.05 \times 10^{-2}$ (volume fraction)
H. $9.00 \times 10^{-3}$ (volume fraction)
I. $1.20 \times 10^{-2}$ (volume fraction)
J. $8.75 \times 10^{-3}$ (volume fraction)
Answer: E
Below 400K, the rate expression for a certain gas-phase reaction is given as:
$$-\frac{\mathrm{d}p_{\mathrm{A}}}{\mathrm{d}t}=3.709\:p_{\mathrm{A}}^{2}\ \mathrm{MPa/h}$$
If the rate equation is expressed as:
$$
(-r_{\mathrm{A}}) \!=\! -\frac{1} {V} \; \frac{\mathrm{d} N_{\mathrm{A}}} {\mathrm{d} t} \!=\! k c_{\mathrm{A}}^{2}
$$
in units of $\mathrm{mol} / (L \cdot h)$, what is the value of the rate constant?

Options:
A. $$
1 0. 8 9 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
B. $$
1 3. 4 2 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
C. $$
1 5. 0 0 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
D. $$
1 2. 3 5 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
E. $$
3 0. 7 0 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
F. $$
8. 9 9 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
G. $$
9 . 8 7 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
H. $$
1 2. 3 4 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
I. $$
2 4 . 5 6 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
J. $$
1 7. 2 5 \mathrm{~ L / ( m o l ~ \cdot~h )}
$$
Answer: H
At 100°C, an esterification reaction is conducted using hydrochloric acid as the catalyst. The reaction equation is as follows:
$$ (A)\text{CH}_3\text{COOH} + (B)\text{C}_2\text{H}_5\text{OH} \overset{k_1}{\operatorname*{\underset{k_2}{\operatorname*{⇌}}}} (P)\text{CH}_3\text{COO}\text{C}_2\text{H}_5 + (S)\text{H}_2\text{O} $$
The reaction kinetics equation is:
$$(-r_\mathrm{A}) = k_1c_\mathrm{A}c_\mathrm{B} - k_2c_\mathrm{P}c_\mathrm{S}$$
Where: $k_1 = 4.76\times10^{-4}\mathrm{~m}^3/($kmol$\cdot$min)
$k_2 = 1.63\times10^{-4}$ m$^3/($kmol$\cdot$min)
If the reaction is carried out in a batch reactor, the feed volume is 0.3785 m³ of aqueous solution, containing 90.8 kg of acetic acid and 181.6 kg of ethanol. The density of the substance system during the reaction is 1043 kg/m$^3$. Calculate the conversion rate of acetic acid at 120 min and its equilibrium conversion rate is ().

Options:
A. 0.359,0.522
B. 0.348,0.560
C. 0.342,0.530
D. 0.357,0.545
E. 0.364,0.549
F. 0.365,0.512
G. 0.340,0.534
H. 0.372,0.538
I. 0.351,0.580
J. 0.355,0.559
Answer: D
The reaction equation for the mixture of hydrochloric acid, octanol, and dodecanol is $$HCl+CH_3(CH_2)_6CH_2OH\xrightarrow{1}CH_3(CH_2)_6CH_2Cl+H_2O$$. This is a parallel reaction, with the reaction rates of octanol (A) and dodecanol (B) expressed as
$$\begin{aligned}(-r_{\mathrm{A}})=&k_{1}c_{\mathrm{A}}c_{\mathrm{C}}\\(-r_{\mathrm{B}})=&k_{2}c_{\mathrm{B}}c_{\mathrm{C}}\end{aligned}$$
where $c_{\mathrm{A}},c_{\mathrm{B}}$, and $c_{\mathrm{C}}$ represent the concentrations of octanol, dodecanol, and hydrochloric acid, respectively. The reaction proceeds under isothermal conditions. The rate constants are given as $$k_{1}=1.6\times10^{-3}\:\mathrm{L/(mol}\cdot\mathrm{min)}\\k_{2}=1.92\times10^{-3}\:\mathrm{L/(mol}\cdot\mathrm{min)}$$. If the initial concentrations are $$\begin{aligned}&c_{\mathrm{AO}}=2.2\:\mathrm{mol/L}\\&c_{\mathrm{BO}}=2\:\mathrm{mol/L}\\&c_{\mathrm{PO}}=1.3\:\mathrm{mol/L}\end{aligned}$$ when the yield of octyl chloride is 34% (calculated based on hydrochloric acid), what is the conversion rate of hydrochloric acid and the yield of dodecyl chloride?

Options:
A. 0.257
B. 0.615
C. 0.478
D. 0.587
E. 0.363
F. 0.342
G. 0.635
H. 0.298
I. 0.128
J. 0.549
Answer: E
In a PFR (Plug Flow Reactor), the oxidative dehydrogenation reaction of butene to produce butadiene is carried out under isothermal conditions at 650°C:
$$
\begin{array} {r} {{(A)\mathrm{C_{4} H_{8} \longrightarrow (P)C_{4} H_{6}+(S)H_{2}}}} \end{array}
$$
Reaction rate equation:
$$
(-r_{\mathrm{A}}) \!=\! k p_{\mathrm{A}} \qquad\mathrm{k \, mol / (m^{3} \cdot h )}
$$
The feed gas is a mixture of butene and steam, with a molar ratio of 1. The operating pressure is 0.10133 MPa, and at 650°C:
$$
k \!=\! 106.48 \qquad\mathrm{k \, mol / (h \cdot m^{3} \cdot MPa )}
$$
The conversion rate of butene is 0.9. Calculate the required space time is ().

Options:
A. $$
12.0s
$$
B. $$
9.7s
$$
C. $$
12.5s
$$
D. $$
11.9s
$$
E. $$
17.6s
$$
F. $$
10.8s
$$
G. $$
14.3s
$$
H. $$
16.4s
$$
I. $$
13.2s
$$
J. $$
15.0s
$$
Answer: I
A first-order irreversible reaction, $\mathrm{A} \longrightarrow \mathrm{P}$, has a reaction rate constant of 3.45 $\mathbf{h}^{-1}$ at 85°C. It is planned to be carried out in a stirred-tank reactor with a volume of 10 $\mathbf{m}^3$. If the final conversion rate is 0.95, the material throughput for this reactor can reach $1.82 \ \mathrm{m^3/h}$. If two stirred-tank reactors with the same volume are connected in series, what would the total volume be?

Options:
A. $$
3.25m^{3}
$$
B. $$
3.64m^{3}
$$
C. $$
4.50m^{3}
$$
D. $$
4.07m^{3}
$$
E. $$
2.89m^{3}
$$
F. $$
3.83m^{3}
$$
G. $$
3.48m^{3}
$$
H. $$
3.01m^{3}
$$
I. $$
5.02m^{3}
$$
J. $$
2.36m^{3}
$$
Answer: B
In a reactor, an isothermal first-order irreversible reaction is carried out. At the reaction temperature, $k=0.05\ min^{-1}$. Using the multi-tank-in-series model method, what is the conversion rate of the reaction?

Options:
A. 0.914
B. 0.763
C. 0.729
D. 0.749
E. 0.678
F. 0.801
G. 0.835
H. 0.882
I. 0.897
J. 0.852
Answer: J
The known liquid-phase reactions are as follows:
$$
\begin{array} {c c}
{{{\mathrm{A + B} \stackrel{1}{\longrightarrow} {P},}}} & {{{r_{\mathrm{P}} = k_{1} c_{\mathrm{A}} c_{\mathrm{B}}}}} \\
{{{2 \mathrm{A} \stackrel{2}{\longrightarrow} {S},}}} & {{{r_{\mathrm{S}} = k_{2} c_{\mathrm{A}}^{2}}}} \\
\end{array}
$$
$$
\frac{k_{1}} {k_{2}} \!=\! 2 \,, \quad c_{\mathrm{A o}} \!=\! c_{\mathrm{B o}}
$$
P is the desired product. Since A is an expensive reactant and difficult to separate from the products, the conversion rate of material A is required to be as high as possible, $x_{\mathrm{A}} = 0.98$. Furthermore, it is desired to obtain a high yield of product P. What is the conversion rate of B in the plug flow reactor?

Options:
A. 0.64
B. 0.61
C. 0.72
D. 0.58
E. 0.66
F. 0.57
G. 0.70
H. 0.55
I. 0.63
J. 0.59
Answer: B
A binary liquid mixture passes through a spherical catalyst bed with a diameter of $d_{\mathrm{P}} = 5 \, \mathrm{mm}$ at an apparent velocity of $0.1 \, \mathrm{m/s}$. The liquid density $\rho$ is $1000 \, \mathrm{kg/m^{3}}$, viscosity $\mu$ is $1 \times 10^{-1} \, \mathrm{Pa \cdot s}$, and the diffusion coefficient $D$ is $8 \times 10^{-10} \, \mathrm{m^{2}/s}$. What is the $S h$?

Options:
A. 478
B. 274
C. 538
D. 512
E. 689
F. 329
G. 425
H. 391
I. 458
J. 346
Answer: I
A second-order irreversible reaction $\mathrm{A} \to \mathrm{B}$ occurs on a porous spherical catalyst with a diameter of 6 mm. When there is no diffusion limitation, the reaction kinetics equation is:
$$
(-r_{\mathrm{A}}) = 0.1 c_{\mathrm{A}}^{2} \quad (\mathrm{mol}/(\mathrm{m}^{3} \cdot \mathrm{s}))
$$
If the concentration of $\mathrm{A}$ in the gas phase is $100 \, \mathrm{mol}/\mathrm{m}^{3}$, the effective diffusion coefficient is $10^{-6} \, \mathrm{m}^{2}/\mathrm{s}$, the reaction proceeds at constant temperature, and the effect of external diffusion can be neglected, what is the internal effectiveness factor of the catalyst?

Options:
A. 0.12
B. 0.15
C. 0.67
D. 0.73
E. 0.50
F. 0.34
G. 0.24
H. 0.45
I. 0.80
J. 0.29
Answer: G
"Roban and Marion" is ________.

Options:
A. The Cosmic Frequency Ode
B. The Galactic Chord Ballad
C. The Universal Spectrum Song
D. The Celestial Range Hymn
E. The Solar Vibration Anthem
F. The Chromatic Harmony Tune
G. The Lunar Scale Symphony
H. The Stellar Arrangement Serenade
I. The Orbital Resonance Chant
J. The Planetary Melody Suite
Answer: C
The characteristic odor of pineapple is due to ethyl butyrate, a compound containing carbon, hydrogen, and oxygen. Combustion of 2.78 mg of ethyl butyrate produces 6.32 mg of CO$_{2}$ and 258 mg of H$_{2}$O. What is the empirical formula of the compound?

Options:
A. $$
\mathrm{C_{2} H_{5} O}
$$
B. $$
\mathrm{C_{5} H_{10} O}
$$
C. $$
\mathrm{C_{3} H_{4} O_{2}}
$$
D. $$
\mathrm{C_{3} H_{6} O}
$$
E. $$
\mathrm{C_{2} H_{6} O}
$$
F. $$
\mathrm{C_{3} H_{8} O}
$$
G. $$
\mathrm{C_{4} H_{8} O_{2}}
$$
H. $$
\mathrm{C_{4} H_{10} O}
$$
I. $$
\mathrm{C_{6} H_{6} O}
$$
J. $$
\mathrm{C_{2} H_{4} O_{2}}
$$
Answer: D
Because atoms are spherical, they cannot occupy all of the space of the cube. The silver atoms pack in the solid in such a way that 74% of the volume of thesolid is actually filled with the silver atoms. Calculate the volume of a single silver atom.

Options:
A. $$
2 6 \, A^{3} \, / \, \mathrm{A g \, a t o m}
$$
B. $$
1 3 \, \text{Å}^{3} \, / \, \text{Ag atom}
$$
C. $$44 \, \text{Å}^3/\text{atom}$$
D. $$52 \, \text{Å}^3/\text{atom}$$
E. $$39 \, \text{Å}^3/\text{atom}$$
F. $$65 \, \text{Å}^3/\text{atom}$$
G. $$81 \, \text{Å}^3/\text{atom}$$
H. $$77 \, \text{Å}^3/\text{atom}$$
I. $$11 \, \text{Å}^3/\text{atom}$$
J. $$98 \, \text{Å}^3/\text{atom}$$
Answer: B
An ideal reversible heat engine with a working substance at an initial state of $350 K, p^\theta$ in 10 mol of water (A) and 293K working between the atmosphere (heat source R). As the temperature of water (A) gradually decreases to $293 K$, it is known that the specific heat capacity at constant pressure of water is $C_{p, m} = 75.3 J \cdot K^{-1} \cdot mol^{-1}$, and it is considered constant.
What are the heat $Q$ absorbed by the atmospheric heat reservoir R and the work $W^{\prime}$ done by the heat engine?

Options:
A. 34.0kJ,3900J
B. 33.5kJ,3650J
C. 37.2kJ,3400J
D. 36.5kJ,3450J
E. 36.0kJ,3500J
F. 35.0kJ,3750J
G. 34.8kJ,3800J
H. 35.5kJ,3702J
I. 38.0kJ,3550J
J. 32.5kJ,3600J
Answer: H
At 273 K, when the pressure increases by 100 kPa, the melting point of ice decreases by $7.42\times10^{-8} K\cdot {Pa}^{-1}$. It is known that the molar volume of ice and water at 273 K and $p^{\Theta}$ are 19.633 cm$^3\cdot\mathrm{mol}^{-1}$ and 18.004 cm$^3\cdot\mathrm{mol}^{-1}$, respectively. What is the molar enthalpy of fusion at 273 K and $p^{\Theta}$?

Options:
A. $$
6 0 5 4 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
B. $$
5 7 2 0 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
C. $$
5 9 0 2 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
D. $$
6 1 0 9 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
E. $$
5 8 1 7 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
F. $$
6 0 0 1 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
G. $$
5 7 5 3 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
H. $$
5 9 9 3 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
I. $$
6 2 3 0 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
J. $$
5 6 8 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
Answer: H
At $273 K$ and $101.3 \times 10^{3} Pa$, the solubility of $O_{2}(g)$ in water is $4.89 cm^{3} / 100 g$ of water, and the solubility of $N_{2}(g)$ is $2.33 cm^{3} / 100 g$ of water. . Assume air is composed of 21% $O_{2}$ and 79% $N_{2}$ (by volume fraction). How much does the freezing point of water saturated with air decrease compared to pure water? (The volume of the dissolved gas is measured at $273 K$ and $p^{\Theta}$)?

Options:
A. 0.00512K
B. 0.00301K
C. 0.00375K
D. 0.00290K
E. 0.00450K
F. 0.00410K
G. 0.00245K
H. 0.00395K
I. 0.00268K
J. 0.00328K
Answer: J
This study investigates the methyl radical combination reaction: $CH_{3} + CH_{3} \longrightarrow C_{2}H_{6}$. Based on viscosity measurements, the molecular diameter of $\mathrm{CH_{4}}$ is $d = 0.414~\mathrm{nm}$, and $d(\mathrm{CH_{4}}) \approx d(\mathrm{CH_{3}})$. What are the collision frequency $Z$ and the half-life $t_{1/2}$ at 273K and a pressure of $0.13~\mathrm{Pa}$?

Options:
A. $$
2. 4 7 \times1 0^{2 2} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 5 8 \times1 0^{-5} \mathrm{s}
$$
B. $$
3. 1 4 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 5 0 \times1 0^{-3} \mathrm{s}
$$
C. $$
2. 7 8 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 2 4 \times1 0^{-4} \mathrm{s}
$$
D. $$
3. 5 5 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 7 7 \times1 0^{-4} \mathrm{s}
$$
E. $$
1. 9 0 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
2. 0 8 \times1 0^{-5} \mathrm{s}
$$
F. $$
2. 6 4 \times1 0^{2 2} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 3 3 \times1 0^{-4} \mathrm{s}
$$
G. $$
3. 7 6 \times1 0^{2 2} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 4 9 \times1 0^{-4} \mathrm{s}
$$
H. $$
3. 8 0 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 6 4 \times1 0^{-3} \mathrm{s}
$$
I. $$
2. 0 3 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
2. 1 1 \times1 0^{-4} \mathrm{s}
$$
J. $$
2. 5 3 \times1 0^{2 3} \mathrm{m^{3}\cdot s^{-1} }
$$
$$
1. 8 9 \times1 0^{-4} \mathrm{s}
$$
Answer: C