a pump is to deliver 150 gpm of water at a head of 120 m. if the pump efficiency is 70%, what is the horsepower rating of motor required to drive the pump?

When you increase the frequency, the number of lines of nodes will increase.

Sound waves create standing waves when they resonate in a confined space. Nodes are the points on a standing wave where there is no displacement or movement. As the frequency increases, the wavelength of the sound wave decreases. This results in more standing waves fitting within the same space. Consequently, the number of lines of nodes increases as more nodes are present due to the increased number of standing waves.

So, when the frequency increases, the number of lines of nodes will increase as well.

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The radial velocity and transit methods are based on the gravitational tug a planet exerts on its star and can be used to determine a planet's mass.

There are several methods that can be used to determine the mass of a planet based on the gravitational tug it exerts on its star. These methods include:

All of these methods are based on the gravitational tug a planet exerts on its star and provide important information about a planet's mass and other characteristics.

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Wavelength shortens, and height increases until it becomes a breaker wave if  wave is stable enough to approach shore.

Wave height and wavelength change when a wave enters shallow water. As the proportion of wave level to frequency, called wave steepness, builds, the wave turns out to be less steady

The waves get closer to the shore before breaking on a moderate slope. Since the water shallows all the more quickly, wave energy is quickly gathered into a little region, so the waves develop exceptionally tall and the peaks twist far forward of the box.

Frequency (f) is the number of complete wavelengths in a given amount of time. The frequency and energy (E) of a wavelength decrease as its size increases. From these situations you might understand that as the recurrence expands, the frequency gets more limited.

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The relationship between the drag forces on the two plates will depend on the specific values of the fluid velocities and densities, as well as the dimensions of the plates. It is not possible to determine the exact relationship between the drag forces without knowing these values.  

When two fluids flow over two identical flat plates with the same laminar free stream velocity, the drag force on each plate is proportional to the fluid velocity difference between the plate and the free stream, the fluid density, and the surface area of the plate.

In this case, if one fluid is twice as dense as the other, it means that it has a higher mass per unit volume than the other fluid. Therefore, it will experience a higher drag force than the other fluid. However, if the two fluids have the same viscosity, then the drag force on each plate will be proportional to the fluid velocity difference between the plate and the free stream and the surface area of the plate.

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A 1206 kg car moving at a velocity of 31 m/s slows down to a velocity of 16 m/s.-18114.87 Ns is the impulse did the car's brakes deliver to the car.

To find the impulse delivered to the car by its brakes, we need to use the formula:
[tex]Impulse = Force * Time[/tex]
Since the car is slowing down, the force exerted by the brakes is in the opposite direction of the car's motion. The force is given by:
[tex]Force =\frac{ (mass of car)  (change in velocity) }{(time)}[/tex]
Here, the mass of the car is given as 1206 kg. The change in velocity is:
Δv = final velocity - initial velocity
   = 16 m/s - 31 m/s
   = -15 m/s
The negative sign indicates that the car is slowing down. We don't know the time it takes for the car to slow down, but we can use another formula to relate the initial and final velocities to the distance traveled and the time taken:
Δx = (initial velocity + final velocity) / 2 x time
Here, Δx is the distance traveled while slowing down. We don't know this distance, but we can assume it's equal to the length of the car, which is about 4 meters. We can rearrange the formula to solve for time:
time = 2 x Δx / (initial velocity + final velocity)
    = 2 x 4 m / (31 m/s + 16 m/s)
    = 0.129
Now we can calculate the force exerted by the brakes:
Force = (mass of car) x (change in velocity) / (time)
     = 1206 kg x (-15 m/s) / 0.129 s
     = -140437 N
Again, the negative sign indicates that the force is in the opposite direction of the car's motion. Finally, we can calculate the impulse delivered by the brakes:
Impulse = Force x Time
       = (-140437 N) x (0.129 s)
       = -18114.87 Ns
The unit for impulse is newton-seconds (Ns), so the answer is approximately -18114.87 Ns. Note that the negative sign indicates that the impulse is in the opposite direction of the car's motion, which is consistent with the force being in the opposite direction as well.

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The total mass of the fission fragments is slightly less than the mass of the original nucleus due to the conversion of some of the mass into energy.

In a fission reaction, the total mass of the fission fragments is slightly less than the mass of the original nucleus. This difference in mass is known as the mass defect, which is converted into energy according to Einstein's famous equation E=mc².

During a fission reaction, a heavy nucleus is split into two lighter nuclei (fission fragments) and some free neutrons. The sum of the masses of the fission fragments and the neutrons is slightly less than the mass of the original nucleus. This is due to the fact that some of the mass is converted into energy in the form of kinetic energy of the fission fragments and neutrons, as well as in the form of gamma rays and other forms of radiation.

The amount of mass that is converted into energy is very small, but because the speed of light (c) is so large, the amount of energy that is released is significant. This is the principle behind nuclear energy and nuclear weapons, which rely on the conversion of a small amount of mass into a large amount of energy.

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Answer:

Magnitude: approximately [tex]1.12 \times 10^{3}\; {\rm N\cdot C^{-1}}[/tex].

Direction: towards the negative point charge.

Explanation:

By Coulomb's Law, at a distance of [tex]r[/tex] from a point charge of magnitude [tex]q[/tex], magnitude of the electric field would be:

[tex]\begin{aligned} E &= \frac{k\, q}{r^{2}}\end{aligned}[/tex],

Where [tex]k \approx 8.99 \times 10^{9}\; {\rm N\cdot m^{2}\cdot C^{-2}}[/tex] is Coulomb's Constant.

In this question, it is given that:

Substitute in the values (note the units) to find the magnitude of the electric field:

[tex]\begin{aligned} E &= \frac{k\, q}{r^{2}} \\ &\approx \frac{(8.99 \times 10^{9})\, (6.8 \times 10^{-6})}{(7.4)^{2}}\; {\rm N\cdot C^{-1}} \\ &\approx 1.12 \times 10^{3}\; {\rm N\cdot C^{-1}}\end{aligned}[/tex].

At a given location, the direction of the electric field would be the same as the direction of the electrostatic force on a positive test charge at that very position.

For example, to find the direction of the electric field in this question, consider a positive test charge placed at the required location.

Charges of opposite signs attract each other. Hence, the hypothetical positive test charge would be attracted to the negative point charge with an electrostatic force pointing towards that negative charge. Direction of the electric field at that position would point in the same direction- towards the negative point charge.

If boy x pushes by straightening his arms out while boy y holds his arms in the original position, the two boys will experience different types of motion. Boy x will experience translational motion, which is the motion of an object moving in a straight line from one point to another. This is because boy x is applying a force to the object in front of him, causing it to move in a straight line.

On the other hand, boy y will experience rotational motion, which is the motion of an object rotating around a fixed axis. This is because boy y is not applying a force in a straight line, but rather is holding the object in its original position. However, if boy x continues to apply force to the object, eventually boy y's arms may rotate as he tries to maintain his grip on the object.

In summary, boy x's motion is translational, as he is pushing the object in a straight line, while boy y's motion is rotational, as he is holding the object in its original position.

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The time it takes for a pendulum to swing to and fro is considered its "period." The period of a pendulum is the time it takes for the pendulum to complete one full oscillation, starting from one extreme position, swinging to the opposite extreme, and returning to the initial position.

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.

So, The time it takes for a pendulum to swing to and fro is considered its "period."

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Careful measurements of the shifting spectral lines in an ordinary star orbiting a black hole can provide valuable information about the presence and properties of the black hole.

By observing the shifting of spectral lines, scientists can infer the presence of a massive object exerting gravitational influence on the star. This gravitational effect, known as gravitational redshift or blueshift, causes the wavelengths of light emitted by the star to shift towards longer or shorter wavelengths, respectively. The careful measurements of these spectral line shifts can provide insights into various aspects of the black hole, such as its mass, spin, and orbital characteristics. The degree of spectral line shift can be used to estimate the gravitational force exerted by the black hole, which in turn helps determine its mass.

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The relative density of paraffin is 0. 9

Relative density can simply be defined as the ratio of the density of a substance to the density of the standard substance

It is also called Specific Gravity.

Relative density is a dimensionless quantity.

Such that If a substance is said to have a relative density less than one then it is less dense compared to a reference substance

The formula for relative density is written as;

R.D = Density of substance/density in water

Substitute the parameters, we have;

R.D = 45/50

Divide the values

R.D = 0. 9

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Aluminum wire is not commonly used in houses anymore, especially for branch circuits. Instead, copper wire is preferred due to its superior conductivity, reliability, and safety.

To start, we need to calculate the cost difference between copper and aluminum wires. We know that copper costs $2.81 per pound and aluminum costs $0.93 per pound. We also know that 2.2 lb equals 1 kg.

Let's assume that we need to use 1000 feet of wire for our house. The weight of the wire will depend on its gauge or thickness, but let's assume it weighs 10 lbs. If we use copper wire, it will cost us 10 lbs x $2.81/lb = $28.10. If we use aluminum wire, it will cost us 10 lbs x $0.93/lb = $9.30.

So, by switching to aluminum wire, we could potentially save $28.10 - $9.30 = $18.80.

We assumed that the resistance of the wires remains constant. In reality, the resistance of aluminum wire is higher than that of copper wire. This means that if we switch to aluminum wire without accounting for the change in resistance, we may experience voltage drops, power losses, and other electrical issues.

To avoid this, we would need to use a thicker gauge of aluminum wire to compensate for its higher resistance. This would increase the weight and cost of the wire, reducing or even eliminating the potential savings.

Moreover, aluminum wire is more prone to corrosion and thermal expansion than copper wire. These factors can lead to increased resistance, loose connections, and fire hazards if not properly addressed.

For these reasons, aluminum wire is not commonly used in houses anymore, especially for branch circuits. Instead, copper wire is preferred due to its superior conductivity, reliability, and safety. However, aluminum wire is still used for some applications, such as service entrance conductors and large feeders, where its cost advantage and lighter weight can outweigh its disadvantages.

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The drag force on the coffee filters can be calculated using this equation, which shows that the drag force is proportional to the velocity squared. This means that as the velocity of the coffee filters increases, the drag force will increase at a faster rate. Therefore, at higher velocities, the coffee filters will experience much larger drag forces than at lower velocities.

The relationship between drag force and velocity for these coffee filters can be described by the drag equation:

FD = (1/2)ρv^2CD A

Where:

FD is the drag force

ρ is the density of the fluid through which the object is moving

v is the velocity of the object relative to the fluid

CD is the drag coefficient, which depends on the shape of the object and its surface properties

A is the cross-sectional area of the object perpendicular to its direction of motion.

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The absolute pressure of a motorcycle tire can be found by adding the gauge pressure to the atmospheric pressure.

Given that the gauge pressure is 255 kPa, we need to determine the atmospheric pressure to calculate the absolute pressure. The standard atmospheric pressure at sea level is approximately 101.3 kPa.Therefore, the absolute pressure of the motorcycle tire can be calculated as follows:
Absolute Pressure = Gauge Pressure + Atmospheric Pressure
Absolute Pressure = 255 kPa + 101.3 kPa
Absolute Pressure = 356.3 kPa
Hence, the absolute pressure of the motorcycle tire is 356.3 kPa.

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The electrode that serves as the anode is sn, while the electrode that serves as the cathode is cu. This means that during the process of electroplating, sn will be oxidized and lose electrons, while cu will be reduced and gain electrons.

It is important to note that the roles of the anode and cathode can switch depending on the specific electrochemical reaction taking place.

In an electrochemical cell, the electrode serving as the anode is Sn, and the electrode serving as the cathode is Cu. This means that Sn is the site of oxidation, losing electrons and forming Sn2+ ions, while Cu is the site of reduction, gaining electrons to form Cu(s) from Cu2+ ions. To sum up, Sn serves as the anode and Cu serves as the cathode in this particular cell.

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When n equal capacitors are connected in series, their total capacitance is reduced because the voltage across each capacitor is shared equally. The equivalent capacitance can be calculated by using the formula:

Ceq = C/ n

Where C is the capacitance of each individual capacitor and n is the total number of capacitors in the series connection.

When the same n capacitors are connected in parallel, their total capacitance is increased because the voltage across each capacitor is the same and their capacitances are additive. The equivalent capacitance can be calculated by using the formula:

Ceq = n x C

Where C is the capacitance of each individual capacitor and n is the total number of capacitors in the parallel connection.

Now, to find the ratio of capacitances in series and parallel arrangements, we can divide the equation for the equivalent capacitance in parallel by the equation for the equivalent capacitance in series:

Ceq (parallel)/Ceq (series) = nC/ C/n = n^2

Therefore, the ratio of capacitances in series and parallel arrangements is 1/n^2. This means that as the number of capacitors in the series or parallel connection increases, the ratio of the equivalent capacitances decreases or increases respectively.

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The potential difference between xi = 1.4 m and xf = 2.4 m can be calculated by subtracting the initial position from the final position. In this case, the potential difference is 1.0 m.

The potential difference, also known as voltage, is a measure of the electric potential energy difference between two points in an electric-field. It represents the work done per unit charge in moving a charge from one point to another. In this scenario, we are given two positions, xi and xf, and we subtract the initial position from the final position to determine the potential difference. The result, 1.0 m, represents the change in potential energy per unit charge between the two points.

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We know that torque is given by the cross product of force and displacement vectors:

τ = r × F

where r is the position vector and × denotes the cross product.

In this case, we are given the torque τ as (11.5 Nm)j, which means that the torque vector has a magnitude of 11.5 Nm and points along the y-axis (since it has only a j-component).

Since the torque is produced by the cross product of r and F, we can choose any value for r as long as it is perpendicular to the y-axis. Let's choose r = xi + zk, where x and z are arbitrary constants.

Taking the cross product of r and F, we get:

τ = (xi + zk) × (fy j)

= -xfy k + zf j

Equating the y-components, we get:

zf = τ

fy

= (11.5 Nm)

fy

Substituting the given value of torque, we get:

zf = (11.5 Nm)

fy

Therefore, fy = zf / (11.5 Nm).

Since we were not given a value for zf, we cannot calculate fy without additional information.

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A frictionless pendulum released from 65 degrees with the vertical will vibrate with the same frequency as if it were released from 5 degrees with the vertical because the period is independent of the amplitude and mass.

The given statement is False.

A force from outside acts on a frictionless pendulum to cause it to move, transferring energy to the pendulum. A frictionless pendulum cannot be regarded as a closed system.

The entropy of the universe will not be impacted by the swing of a frictionless pendulum. This is so that a frictionless pendulum can move from its initial position to its final state through a reversible thermodynamic process.

The resonant mechanism supporting this type of pendulum has a single resonant frequency. Frequency is the measure of how many oscillations occur in a second. F or n is used to indicate it.

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The time it takes for photoelectrons with a work function of 2.7 eV and ejected by photons with a wavelength of 420 nm to travel 2.50 cm to a detection device is approximately 1.17 x 10⁻⁷ seconds.

When a photon with energy greater than the work function of a metal strikes the metal surface, it can eject an electron from the metal. The ejected electron, known as a photoelectron, carries kinetic energy equal to the difference between the photon energy and the work function of the metal.

In this problem, the work function of the metal is given as 2.7 eV and the wavelength of the photon is 420 nm. Using the relationship between photon energy and wavelength (E=hc/λ), we can calculate the energy of the photon as 2.95 eV. Therefore, the kinetic energy of the photoelectrons is 0.25 eV.

To calculate the time it takes for the photoelectrons to travel 2.50 cm to the detection device, we need to use the equations of motion. The distance traveled by the electrons is equal to the product of their velocity and the time of flight. The velocity of the electrons can be calculated using their kinetic energy and the equation for the kinetic energy of a particle (K = 1/2mv²). Solving for v, we get v = √(2K/m), where m is the mass of the electron.

Using the mass of the electron and the kinetic energy calculated earlier, we can determine the velocity of the photoelectrons. Then, using the distance traveled (2.50 cm) and the velocity, we can calculate the time of flight. The final answer is approximately 1.17 x 10⁻⁷ seconds.

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Complete Question:

Photoelectrons from a metal with a work function of 2. 7 ev are ejected by photons with a wavelength of 420 nm. How long does it take these electrons to travel 2.50 cm to a detection device?

If Larry stops selling fruits and vegetables, a food desert will be created in the town.

A food desert is an area, typically in low-income communities, where there is limited access to affordable and nutritious food. In this case, without Larry's store providing fresh fruits and vegetables, the residents will face challenges in obtaining these essential food items.

The people in town are more likely to experience the following consequences:

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The frequency of photons with an energy of 4.0 eV is approximately 9.64 × 10^14 Hz.

The energy of a photon is related to its frequency by the equation:

E = hν

where E is the energy of the photon, h is Planck's constant, and ν (nu) is the frequency of the photon. We can use this equation to find the frequency of photons with an energy of 4.0 eV.

First, we need to convert the energy of the photon from electron volts (eV) to joules (J). We know that 1 eV is equal to 1.6 × 10^-19 J, so:

E = 4.0 eV × 1.6 × 10^-19 J/eV = 6.4 × 10^-19 J

Now we can rearrange the equation E = hν to solve for the frequency ν

ν = E / h

where h is Planck's constant, which has a value of 6.626 × 10^-34 J·s.

ν = (6.4 × 10^-19 J) / (6.626 × 10^-34 J·s) ≈ 9.64 × 10^14 Hz

Therefore, the frequency of photons with an energy of 4.0 eV is approximately 9.64 × 10^14 Hz. This represents the frequency of electromagnetic radiation necessary to break the bond in the molecule, since photons of this frequency carry enough energy to overcome the bond's binding energy and dissociate the molecule.

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The electric current i in the given series circuit containing three different resistors and a voltage source is 0.695 A.

To calculate the electric current i in a series circuit containing three different resistors (r₁ = 39 ω, r₂ = 44 ω, and r₃ = 32 ω) and a voltage source (v = 80.0 v), we need to use Ohm's Law and the concept of total resistance.

First, we need to find the total resistance of the circuit by adding up the individual resistances. Therefore, the total resistance (R) of the circuit is:

R = r₁ + r₂ + r₃
 = 39 ω + 44 ω + 32 ω
 = 115 ω

Next, we can use Ohm's Law (V = IR) to find the electric current i flowing through the circuit. Rearranging the formula, we get:

I = V / R
 = 80.0 v / 115 ω
 = 0.695 A

Therefore, the electric current i flowing through the circuit is 0.695 A.

In summary, the electric current i in the given series circuit containing three different resistors and a voltage source is 0.695 A.

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For a series circuit containing three different resistors, r1 = 39 ω, r2 = 44 ω, and r3 = 32 ω, and a voltage source that provides v = 80.0 v, then the electric current flowing in the circuit is approximately 0.696 A.

To find the current flowing in the circuit, we need to apply Ohm's law and the rules for series circuits

1. In a series circuit, the same current flows through all the resistors.

2. The total resistance in a series circuit is the sum of the individual resistances.

Using Ohm's law, we have

V = IR

Where V is the voltage of the source, I is the current flowing in the circuit, and R is the total resistance.

To find the total resistance, we add the individual resistances

R = R1 + R2 + R3 = 39 Ω + 44 Ω + 32 Ω = 115 Ω

Now we can solve for the current I

I = V / R = 80.0 V / 115 Ω ≈ 0.696 A

Therefore, the electric current flowing in the circuit is approximately 0.696 A.

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When determining whether an object is transparent or opaque, we are essentially assessing its ability to transmit light. A transparent material allows light to pass through it easily, whereas an opaque material does not.
For example, a clear glass window is transparent as it allows light to pass through it, while a brick wall is opaque as it does not allow light to pass through it. Similarly, a piece of cling film or saran wrap is transparent as it allows light to pass through it, whereas aluminum foil is opaque as it does not allow light to pass through it.
Answering more than 100 words, some other examples of transparent materials include glass, water, and certain types of plastics. On the other hand, some examples of opaque materials include wood, metals, and most types of stone.
It's important to note that some materials can be partially transparent, meaning that they allow some light to pass through but not all. For example, frosted glass or wax paper are semi-transparent, as they allow some light to pass through but not as much as clear glass or cling film.
In summary, whether an item is composed of transparent or opaque material depends on its ability to transmit light. Transparent materials allow light to pass through them, while opaque materials do not.
When discussing materials, they can be classified as transparent or opaque based on their ability to transmit light. Transparent materials, like glass and clear plastic, allow light to pass through them with little to no distortion. On the other hand, opaque materials, such as wood or metal, block the transmission of light.
For example, a glass window is a transparent material, as it allows light to pass through easily, enabling you to see clearly outside. Conversely, a wooden door is an opaque structure, as it does not allow light to pass through and you cannot see through it.
Remember to always consider the properties of each item when determining if it's transparent or opaque in relation to light transmission.

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The key thing to remember is that transparency and opacity are determined by a material's ability to allow light to pass through it. By considering the specific properties of an item, you can determine whether it is composed of a transparent or opaque material.

When it comes to determining whether an item is composed of transparent (clear) material or is an opaque structure not involved in the transmission of light, there are a few things to consider.First, it's important to understand the difference between transparency and opacity. Transparency refers to the ability of a material to allow light to pass through it, while opacity refers to the opposite - the ability of a material to block or absorb light.

With that in mind, here are some examples of items and whether they are composed of transparent or opaque materials:
- Window glass: transparent (clear) - window glass is specifically designed to allow light to pass through it, so it is a great example of a transparent material.
- Brick wall: opaque - brick is not designed to allow light to pass through it, so it is considered an opaque structure.
- Plastic water bottle: transparent (clear) - many plastic water bottles are made from a clear plastic that allows you to see the water inside, making it a transparent material.
- Wooden desk: opaque - wood is a solid material that does not allow light to pass through it, so a wooden desk would be considered an opaque structure.
- Sunglasses: partially transparent - sunglasses are often made with lenses that are designed to block some light while allowing other wavelengths to pass through, making them a partially transparent material.

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Answer:

the work done by the tugboat was 550,800,000 J and the distance that the cargo ship was pulled was 1,800 m

Explanation:

To solve this problem, we need to use the following formulas:

Work done = Power x Time

Power = Force x Velocity

Work done = Force x Distance

a. To find the work done by the tugboat, we can use the first formula and plug in the given values:

Work done = Power x Time Work done = 153,000 W x 3,600 s Work done = 550,800,000 J

b. To find the distance that the cargo ship was pulled, we can use the third formula and rearrange it to solve for distance:

Work done = Force x Distance Distance = Work done / Force Distance = 550,800,000 J / 306,000 N Distance = 1,800 m

The width of the river is approximately 329.9 meters.

We want to find the width of the river, which is represented by x in the diagram. We know the distance from point C to point A is 275 m, and we know the angle CAB is 57° 28′.

First, we need to find the distance from point C to point B. We can use the tangent function:

tan(θ) = opposite / adjacent

tan(57° 28′) = CB / 275

CB = 275 * tan(57° 28′)

CB ≈ 408.3 m

Now we can find the width of the river x using the sine function:

sin(θ) = opposite / hypotenuse

sin(57° 28′) = x / 408.3

x = 408.3 * sin(57° 28′)

x ≈ 329.9 m

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Shifts in binding energy of photoelectron peaks in XPS provide information on elemental composition, oxidation state, and chemical bonding of materials, aiding in surface analysis and identifying chemical environments.

Shifts in the binding energy of photoelectron peaks obtained from X-ray photoelectron spectroscopy (XPS) are indicative of various properties of materials. The technique enables the determination of elemental composition by comparing the measured binding energies to reference data. Additionally, the shifts provide insights into the oxidation state of elements, allowing for the analysis of chemical transformations and reactions. Furthermore, variations in binding energy can be attributed to different chemical environments, revealing details about the chemical bonding and molecular structure of the material. XPS is surface-sensitive, making it ideal for investigating surface reactions, adsorption phenomena, and the presence of contaminants. By analyzing the binding energy shifts, researchers can gain a deeper understanding of the electronic structure, surface properties, and reactivity of a wide range of materials, encompassing metals, semiconductors, polymers, and biomaterials.

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2.79 x 10⁻⁷ m is the wavelength of light with an energy of 427 kj/mol  by Avogadro's number

To determine the wavelength of light with an energy of 427 kJ/mol, we can use the formula E=hc/λ, where E is the energy of the light, h is Planck's constant (6.626 x 10⁻³⁴ J.s), c is the speed of light (2.998 x 10⁸ m/s), and λ is the wavelength of the light.
First, we need to convert the energy from kJ/mol to J/photon by dividing it by Avogadro's number (6.022 x 10²³) to get the energy per photon.
427 kJ/mol / (6.022 x 10²³ photons/mol) = 7.10 x 10⁻¹⁹ J/photon
Now we can plug this value into the formula and solve for λ:
7.10 x 10⁻¹⁹ J/photon = (6.626 x 10⁻³⁴ J.s)(2.998 x 10^8 m/s) / λ
λ = (6.626 x 10⁻³⁴ J.s)(2.998 x 10⁸ m/s) / 7.10 x 10⁻¹⁹ J/photon
λ = 2.79 x 10⁻⁷ m
Therefore, the wavelength of light with an energy of 427 kJ/mol is approximately 2.79 x 10⁻⁷ m.

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The ratio of their de Broglie wavelengths is approximately the inverse of the ratio of their momenta, or 1:1836. Therefore, the electron's de Broglie wavelength is about 1836 times greater than that of a proton moving at the same speed.

The de Broglie wavelength of a particle is given by the equation λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. The momentum of a particle is given by p = mv, where m is the mass of the particle and v is its velocity. Since the electron and proton have the same speed, their momenta will be in the ratio of their masses. The mass of an electron is approximately 1/1836 times that of a proton. Therefore, the ratio of their momenta is approximately 1836:1. Thus, the ratio of their de Broglie wavelengths is approximately the inverse of the ratio of their momenta, or 1:1836. Therefore, the electron's de Broglie wavelength is about 1836 times greater than that of a proton moving at the same speed.

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