If the velocity of water spinning a turbine is increased by a factor of 4, the maximum energy, per kilogram of water, from the turbine will increase by a factor of 4.
If the velocity of water spinning a turbine is increased by a factor of 4, the maximum energy, per kilogram of water, from the turbine will increase by a factor of 16.Explanation:The rate at which work is done or energy is transferred is called power. P = F × v is the formula for power, where P is power, F is force, and v is velocity.
The maximum energy that can be obtained by a turbine from a kilogram of water is proportional to the square of the velocity of the water. That is, if we double the velocity of the water going through a turbine, we obtain four times as much energy per kilogram of water passing through the turbine.
To illustrate this, we can use the formula for kinetic energy KE = 1/2mv² where KE is kinetic energy, m is mass, and v is velocity. Because kinetic energy is proportional to the square of the velocity, if we double the velocity, we quadruple the kinetic energy.
The kinetic energy is multiplied by three when the velocity is tripled, and so on. Therefore, if the speed of water spinning a turbine is raised by a factor of 4, the maximum energy produced by the turbine will rise by a factor of 16 per kilogramme of water.
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(q020) the conventional system of signs that indicate relative durations of long and short sounds is called
The term "musical notation" refers to the widespread symbology used to indicate how far apart long and short sounds should be from one another.
What is fundamental musical notation?a collection of visual instructions for performing music, a visual representation of a heard or imagined musical sound, or musical notation. It usually takes the form of writing or printing and is a deliberate, comparatively time-consuming activity. One of two purposes—as a memory aid or for communication—causes its use.
What are the twelve notes in music?Typically, Western music employs 12 notes: C, D, E, F, G, A, B, plus five flats and comparable sharps in between: C sharp/D flat (they are the same note, simply referred to differently depending on the context).
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if a girl carries 10kg bag
Jaune moves North 15 meters, South 5 meters, then North 8 meters. How far is Jaune DISPLACED?
Answer:
Displacement = 7 meters
Explanation:
Given that,
Jaune moves North 15 meters, South 5 meters, then North 8 meters. We need to find the displacement of Jaune.
The attached figure shows the motion of Jaune.
We know that, Displacement = final position - initial position
Initially, Jaune traveled 15 meters due North and finally, he travel 8 meters North.
Displcement = 15 m - 8 m
Displacement = 7 m
So, the displacement of Jaune is 7 meters.
A horse running at a velocity of 10 m/sec[W] accelerates at a rate of 6. 25 m/sec 2 [W] for 2 sec. What is the final velocity of the horse?
The final velocity of the horse is 22.5 m/s [W]. Velocity is a vector quantity that describes the rate at which an object changes its position. It includes both magnitude (speed) and direction.
To find the final velocity of the horse, we can use the formula for calculating velocity:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Given:
Initial velocity (u) = 10 m/s [W]
Acceleration (a) = 6.25 m/s^2 [W]
Time (t) = 2 sec
Substituting the given values into the formula, we can calculate the final velocity:
v = 10 m/s + (6.25 m/s^2)(2 sec)
v = 10 m/s + 12.5 m/s
v = 22.5 m/s
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seawater velocity = 1478 m/s water depth = 509 m sandstone velocity = 2793 m/s thickness=1003 m mudstone velocity= 2240 m/s thickness = 373 m Air Gun Energy Source Note: Illustration is not to scale. Hydrophone Receivers seafloor sand/mud 2. In the marine seismic acquisition example shown, you are interested in two events observed in the seismic trace that is recorded at the first hydrophone. One is a first-order multiple (double bounce) off the seafloor. The other is a primary reflection from the sand/mud interface for which the energy ray-path has a takeoff angle of 9 degrees from vertical as shown. Assume horizontal rock layers and isotropic velocities. Which of the two events arrives at the hydrophone first-the primary or the multiple? Clearly show your calculations and include a simple drawing of the two- event seismic trace. 3. How long does it take for energy to travel directly from the air gun to the first hydrophone (no bounces)? 4. What is the maximum takeoff angle at which seismic energy can reflect from the sand/mud interface? Explain what happens to the energy for larger angles. 5. Explain the relative direction of travel for energy that is transmitted into the mudstone.
2. The primary reflection from the sand/mud interface will arrive first at the hydrophone. To determine which event arrives first, we need to calculate the two-way travel times (TWTT) for each event. The TWTT for the primary reflection from the sand/mud interface is:
TWTT = (2 × depth × sin (angle of incidence)) / velocity
TWTT = (2 × 509 × sin (9)) / 1478TWTT = 0.317 s
The TWTT for the double bounce off the seafloor is:TWTT = (2 × depth) / velocityTWTT = (2 × 509) / 1478TWTT = 0.689 s
Therefore, the primary reflection arrives first at the hydrophone. Here is a simple drawing of the two-event seismic trace:
3. To calculate the time it takes for energy to travel directly from the air gun to the first hydrophone, we need to determine the distance between them and divide it by the velocity of sound in seawater. Using the given values, we have:
Distance = depth + (thickness of sand/mud) + (thickness of mudstone)
Distance = 509 + 1003 + 373
Distance = 1885 m
Velocity of sound in seawater = 1478 m/s
Time = Distance / VelocityTime = 1885 / 1478Time = 1.276 s
Therefore, it takes 1.276 seconds for energy to travel directly from the air gun to the first hydrophone.
4. The maximum takeoff angle at which seismic energy can reflect from the sand/mud interface is called the critical angle. This angle can be calculated using Snell's law:
n1 × sin (angle of incidence) = n2 × sin (angle of refraction)
where n1 and n2 are the velocities of the two materials and the angle of refraction is 90 degrees (since seismic energy travels along a horizontal path once it reaches the interface).
For the sand/mud interface, the critical angle is:
n1 × sin (critical angle) = n2 × sin (90)n1 / n2 = cos (critical angle)critical angle = cos^-1 (n1 / n2)
Using the given values:
n1 = 2793 m/s (sandstone velocity)n2 = 2240 m/s (mudstone velocity)critical angle = cos^-1 (2793 / 2240)
critical angle = 35.9 degrees
Seismic energy cannot reflect from the sand/mud interface at angles greater than the critical angle. For larger angles, the energy will be transmitted into the mudstone.
5. When seismic energy is transmitted into the mudstone, it travels in all directions away from the source. However, the energy will be attenuated (reduced in amplitude) as it travels through the mudstone due to its relatively low velocity compared to the sandstone and seawater.
As a result, the mudstone acts as a barrier that blocks or reduces the energy that would otherwise be transmitted deeper into the subsurface.
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which of the following includes an example of a chemical property of an element?
a. aluminum is solid at room temperature
b. sulfur is not shiny and is not malleable
c. sodium is a solid at room temperature and reacts to other elements.
d. silicone is shiny and is a poor conducted of electricity
Answer:
d. silicone is shiny and is a poor conducted of electricity
The best option that describes an example of a chemical property of an element is silicone is shiny and is a poor conductor of electricity. Option D is correct.
The chemical properties of elements are those properties that are noticeable during the course of a chemical reaction.
Aluminum is solid but not at room temperature. The elements that are solid at room temperature are iron, zinc, gold, and silver. Hence option is incorrect. The same is applicable to options B and C.
The best option that describes an example of a chemical property of an element is silicone is shiny and is a poor conductor of electricity
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Which type of circuit would be best to use for lights used for decorations?
Answer:
led would be the best for decorations
Explanation:
Can you please help me it’s due at 11:59 please
Answer:
1 is summer, 2 is most likely winter,3 is most likely summer, and 4 is fall
Explanation:
Questions
2.
Rewrite the following quantities using suitable
prefixes
5000 000
At 0°C, the specific volumes of saturated solid water (ice) and saturated liquid water are, respectively, v₁ = 1.0911 x 10-3 m³/kg and v,= 1.0002 x 10-3 m³/kg, and the change in specific enthalpy on melting is h=333.4 kJ/kg. Assuming her and vir= VV, do not vary significantly with temperature, estimate the melting temperature of ice at (a) 250 bar, (b) 500 bar. Locate your answers on a sketch of the p-T diagram for water.
The melting temperature of ice at different pressures can be estimated using the Clausius-Clapeyron equation, which relates the change in temperature with pressure for a phase change. The equation is given by:
ln(P₂/P₁) = ΔHvap/R * (1/T₁ - 1/T₂)
Where:
- P₁ and P₂ are the initial and final pressures respectively.
- ΔHvap is the enthalpy of vaporization.
- R is the gas constant.
- T₁ and T₂ are the initial and final temperatures respectively.
In this case, we want to estimate the melting temperature of ice at two different pressures, 250 bar and 500 bar. The initial pressure is atmospheric pressure (1 bar) and the initial temperature is 0°C.
(a) For 250 bar:
Using the Clausius-Clapeyron equation, we have:
ln(250/1) = (333.4*10³)/(8.314) * (1/(273.15) - 1/T₂)
ln(250) = 40,107/T₂ - 40,107/273.15
ln(250) + 40,107/273.15 = 40,107/T₂
T₂ = 40,107 / (ln(250) + 40,107/273.15)
(b) For 500 bar:
Using the same equation, we have:
ln(500/1) = (333.4*10³)/(8.314) * (1/(273.15) - 1/T₂)
ln(500) = 40,107/T₂ - 40,107/273.15
ln(500) + 40,107/273.15 = 40,107/T₂
T₂ = 40,107 / (ln(500) + 40,107/273.15)
To locate the answers on a sketch of the p-T diagram for water, you would need to plot the points (250 bar, T₂) and (500 bar, T₂) on the diagram.
Please note that these calculations are based on assumptions such as constant heat capacities and neglecting the variation of volume with temperature. Also, double-check the units and conversion factors in the calculations to ensure accuracy.
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Sodium and potassium are in the same group in the periodic table. Sodium is in the 11th position. How many valence electrons does potassium have?
which term describes weathering?breaks down rockcauses heating and coolingmakes pebbles and rocks largercarries away fragments
The term that describes weathering breaks down the rock. Weathering is the breakdown of rocks, soil, and other geological material under the influence of physical, chemical, or biological agents.
The term weathering refers to a variety of processes that wear away or weaken rocks over time. Weathering is classified into two categories: chemical and physical. Chemical weathering occurs when rock is chemically broken down by reactions with acid rain or other chemicals.
Physical weathering, on the other hand, occurs when rock is broken down into smaller pieces by water, wind, ice, or other forces that do not change the chemical composition of the rock. In conclusion, the term that describes weathering is breaking down the rock.
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A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110.0 m.
Determine the acceleration of the car.
Answer: a = 8.10 m/s^2
Explanation:
The car starts at rest, so:
V0= 0m/s
t= 5.21m/s
S = 110.0 m
Using one of the five uniformly accelerated motion formulas
S= V0t + 1/2 at^2
but V0 = 0 m/s
S = 1/2 at^2
a= 2S/ t^2
a = 2(110m)/ (5.21 s)^2
a = 8.10 m/s^2
A particle of mass 1 MeV/c2 and kinetic energy 2 MeV collides with a staionary particle of mass 2 MeV/c2. After the collision, the particles stick together. Find:
a) the speed of the first particle before the collision
b) the total energy of the first particle before the collision
c) the initial total momentum of the system
d) the total kinetic energy after the collision
e) the mass of the system after the collision
The speed of the first particle before the collision is 2 m/s. The total energy of the first particle before the collision is 4.64 x 10^-13 J.
(1/2)mv² = 3.2 x \(10^{-13\) J
v² = (2 x 3.2 x\(10^{-13\)J) / (1 MeV/c²)
v² = 6.4 x\(10^{-13\) J / (1 MeV/c²)
v² = 6.4 x \(10^{-13\) J / (1.6 x \(10^{-13\) J)
v² = 4
v = √4 = 2 m/s
E = (1 MeV/c²)(3 x \(10^8\) m/s)²
E = (1 x 1.6 x \(10^{-13\) J)(9 x\(10 ^{16\)m²/s²)
E = 1.44 x \(10^{-13\)J
The total energy of the first particle before the collision is the sum of the kinetic energy and the rest energy:
Total energy = Kinetic energy + Rest energy
Total energy = 3.2 x\(10^{-13\)J + 1.44 x \(10^{-13\) J
Total energy = 4.64 x\(10^{-13\) J
Collision refers to the interaction between two or more objects that exert forces on each other for a brief period of time. It is a fundamental concept used to understand the behavior of particles and objects in motion.
During a collision, the objects involved experience a change in their velocities and sometimes their shapes. Collisions can be categorized into two types: elastic and inelastic. In an elastic collision, kinetic energy and momentum are conserved, meaning that the total energy and momentum before the collision are equal to the total energy and momentum after the collision. In an inelastic collision, kinetic energy may be lost due to deformation or the formation of new objects, but momentum is still conserved.
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Complete Question:
A particle of mass 1 MeV/c2 and kinetic energy 2 MeV collides with a stationary particle of mass 2 MeV/c2. After the collision, the particles stick together. Find:
a) the speed of the first particle before the collision
b) the total energy of the first particle before the collision
if the current is 3amps and resistance is 10 ohms what is the voltage?
If the current is 3amps and resistance is 10 ohms, the voltage is 30 V.
Ohm's Law states that voltage is equal to the product of current and resistance.
Using the formula V=IR, where V is voltage, I is current and R is resistance, we can calculate the voltage in this scenario.
So, if the current is 3 amps and resistance is 10 ohms, we can plug in these values in the formula: V= 3A x 10Ω V= 30 volts
Therefore, the voltage in this scenario is 30 volts.
This means that in order to maintain a current of 3 amps through a resistance of 10 ohms, a voltage of 30 volts is required.
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What happens during heat transfer within Earth?
Group of answer choices
Warmer, less dense material rises.
Explanation:
This happens when heat transfers.
The capacitance of a parallel plate capacitor is _________.
Group of answer choices
proportional to charge
proportional to the plate separation
proportional to the area of the plates
proportional to the potential difference
None of the above
The capacitance of a parallel plate capacitor is proportional to the area of the plates. Hence option C is correct.
A capacitor is a device that stores electrical energy in an electric field by collecting electric charges on two isolated surfaces. It is a passive electronic component with two terminals.
Capacitance is the effect of a capacitor. While any two electrical conductors in close proximity in a circuit have some capacitance, a capacitor is a component designed to provide capacitance to a circuit. The capacitor was initially referred to as a condenser, a word that is still used in a few compound names, such as the condenser microphone.
Capacitance of the capacitor is directly proprtional to the area of the plates and inversly proprtional to the distance between them.
C = ∈A/d
Hence option C is correct.
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what current (in ka) is needed to transmit 160 mw of power at a voltage of 20.0 kv?
The current needed to transmit 160 MW of power at a voltage of 20.0 kV is 8,000 A (amperes) or 8 kA (kiloamperes).
To calculate the current needed to transmit 160 MW (megawatts) of power at a voltage of 20.0 kV (kilovolts), we can use the formula for power:
Power (P) = Voltage (V) × Current (I)
Rearranging the formula to solve for current, we get:
Current (I) = Power (P) / Voltage (V)
Given that the power is 160 MW and the voltage is 20.0 kV, we can plug in these values to calculate the current:
Power (P) = 160 MW = 160 × 10⁶ W (since 1 MW = 10⁶ W)
Voltage (V) = 20.0 kV = 20.0 × 10³ V (since 1 kV = 10³ V)
Substituting these values into the formula, we get:
Current (I) = 160 × 10⁶ W / 20.0 × 10³ V
Simplifying, we get:
Current (I) = 8,000 A
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BRAINLIEST : A car starts from rest and accelerates at a rate of 40 m/s2 over a time if 2.4 s. How fast is the car going at the end of this time interval?
Please explain
Answer:
\(\boxed {\boxed {\sf 96 \ m/s}}\)
Explanation:
We are asked to find the final velocity of the car given the acceleration and time. We can use the following kinematics equation to calculate the final velocity.
\(v_f=v_i+(a \times t)\)
The car starts from rest, so the initial velocity is 0 meters per second. It accelerates a rate of 40 meters per square second over a period of time of 2.4 seconds.
\(v_i\)= 0 m/s a= 40 m/s²t= 2.4 sSubstitute the values into the formula.
\(v_f= 0 \ m/s + ( 40 \ m/s^2 \times2.4 \ s)\)
Solve inside the parentheses.
40 m/s/s * 2.4 s = 40 m/s * 2.4=96 m/s\(v_f= 0 \ m/s + (96 \ m/s)\)
Add.
\(v_f= 96 \ m/s\)
The final velocity of the car is 96 meters per second.
Two spaceships orbit a planet. Spaceship A orbits at a distancer from the centre of the planet. Spaceship B orbits at a distance 2r from the centre of the planet. What is the orbital speed of A compared with B?
The orbital speed of spacecraft A is one-half that of craft B.
What is a spacecraft's orbital speed?A spacecraft needs to move at a speed of roughly 8 kilometres (5 miles) per second to maintain a circular orbit just above the Earth's atmosphere. A faster launch velocity will cause the spacecraft to swing farther away from Earth. If it is increased to 11 km/s (7 miles/s), the spaceship will completely depart the planet.
A satellite's orbital speed is influenced by the mass of the planet and how far it is from the planet's centre. The third law of Kepler states that the square of a satellite's orbital period is proportionate to the cube of the distance of the satellite from the planet's centre:
T is the orbital period, and r is the separation from the planet's centre (T2 r3).
To calculate the ratio of their orbital speeds, we can apply the equation above:
Spacecraft A and B's orbital speeds are given by v_A and v_B, respectively, while their distances from the planet's centre are given by r_A and r_B, respectively.
r_B = 2r_A
This is substituted into the equation above to produce the result: v_A / v_B = r_B / r_A = 2.
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Which of the following best compares the atom of an element to an ion and an isotope of this element? *
1 point
An atom and its ion have the same charge but different masses. The isotope of an atom will have the same mass.
An atom, its ion, and its isotope will all have the same mass. The isotope of an atom will have a different charge.
The ion of an atom will have a different mass but the same charge. The isotope of an atom will have a different charge.
The ion of an atom will have a different charge but the same mass. The isotope of an atom will have a different mass.
Answer:
the ion of an atom will have a different charge but the same mass.The isotope of an atom will have a different mass
The statement that compares the atom of an element to an ion and isotope is: the ion of an atom will have a different charge but the same mass. The isotope of an atom will have a different mass.
WHAT IS AN ION AND ISOTOPE:An ion is a charged atom while an isotope is the atoms of an element with the same atomic number but different atomic mass. For example, Mg2+ is the ionic form of Magnesium while C-12 and C-14 are isotopes of carbon.Therefore, the statement that compares the atom of an element to an ion and isotope is: the ion of an atom will have a different charge but the same mass. The isotope of an atom will have a different mass.
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TUOI 7. A stone dropped from a window reaches the ground in 1.5 seconds. Calculate the height of the window above the ground
The height of the window above the ground is 11.025m
The first step is to write out the parameters given in the question
U(initial velocity)= 0
Time= 1.5 seconds
Acceleration due to gravity= 9.8
Therefore the height of the window can be calculated as follows
S= ut + 1/2gt²
= 0(1.5) + 1/2(9.8)(1.5²)
= 0 + 1/2(9.8)(2.25)
= 1/2(22.05)
= 0.5×22.05
= 11.025
Hence the height of the window above the ground is 11.025m
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anne is working on a research project that involves the use of a centrifuge. her samples must first experience an acceleration of 100???? , but then, the acceleration must increase by a factor of eleven. by how much will the rotational speed have to increase? express your answer as a fraction of the initial rotation rate.
if the final acceleration increases by a factor of 11, then the final rotation should be 3.32\(w_{1}\) times the initial rotation.
The relation between the centripetal acceleration and rotation speed of the particle is given as
a=rω²
Where ω is the angular speed and r radius of the circle
Now, rearranging the above equation we will get
ω = √\(\frac{a}{r}\)
The initial acceleration is given by
a₁ =100g
But then, the acceleration increases by a factor of 11. Which means the final acceleration of the particle is
a₂ = 11 (100g)
= 1100g
The final and initial rotation speeds are related as
\(\frac{w1}{w2}\) = √\(\frac{a2}{a1}\)
\(w_{2}\) = √\(\frac{1100g}{100g}\) x \(w_{1}\)
\(w_{2}\) = 3.32 \(w_{1}\)
The final rotation should be 3.32\(w_{1}\) times the initial rotation.
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1. A car increases its speed from 9.6 meters per second to 12.8 meters per second in 8.0
seconds. What is the acceleration of the car?
a. 2.4 m/s²
b. 5.2 m/s²
c. .4 m/s²
d. 2.8 m/s²
Answer:
c
Explanation:
Acceleration = change in speed / change in time
= (12.8- 9.6) / ( 8) = 3.2 / 8 = .4 m/s^2
Among discovered meteorites, we have found some with all the following origins EXCEPT... A. Being a fragment of a shattered asteroid. B. Being a fragment from the surface of the Moon. C. Being a fragment from Cornet Halley. D. Being a fragment from the surface of Mars.
We have discovered meteorites with ALL of the following sources, EXCEPT one being a piece of the comet Halley, among the identified meteorites.
Where do most meteors come from? Most meteors come from what place?Our solar system is where all meteorites originate. Most of them are pieces of ancient asteroids that shattered into smaller pieces in the asteroid belt, which is situated between Mars and Jupiter.
How did asteroids get their start?Asteroids are remains of the solar system's creation, which occurred around 4.6 billion years ago. Early on, the formation of Jupiter stopped any planetary bodies from forming in the space between Mars and Jupiter, causing the small particles that were already present to smash and shatter into the asteroids we know today.
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A student is playing with a pendulum. He gives the ball a push and watches the ball as it swings. After a while, the ball stops swinging.Why does the ball stop swinging?
In a series-parallel circuit, you can find the total
resistance of the circuit by calculating the resistance of
each parallel section and then adding the series
resistances to these values to obtain a single
resistance. (This class is about cars )
If you know the total current and voltage across the circuit, then you can find the total resistance using the Ohm's Law, that is : R = V / I.
How is resistance calculated in series-parallel circuit?In a series-parallel circuit, one can find the total resistance of circuit by calculating resistance of each parallel section and then adding reciprocals of these values to obtain total resistance of parallel section. Next, you can add series resistances to this total resistance to obtain single resistance for entire circuit.
In series the total resistance is equal to the sum of the resistors. In parallel, inverse of the total resistance is equal to the the sum of the inverse of each individual resistor.
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Note: The question given on the portal is incomplete. Here is the complete question.
Question: How is resistance calculated in a series-parallel circuit?
Fast cooling rates tends to promote transformations that result in relatively course structures (e.g. for a reaction resulting in multiple product phases, the secondary phase particles are (i) fewer in count (ii) larger in size and (iii) relatively spherical). True or False
For most materials, the formation of a glass from a liquid typically requires extremely slow cooling rates, thereby taking on the order of years or even centuries to produce. True or False
For a system below its melting point, a glass (or non-crystalline / amorphous) configuration is the most energetically stable state that is possible, since it offers the lowest Gibbs free energy for all possible solid configurations. True or False
When measured experimentally, most materials tend to exhibit the same exact temperature value for both melting and freezing (crystallization). True or False
Gray iron is a microstructure that tends to result when a cast iron of appropriate composition is cooled fairly quickly. True or False
Coarsening refers to the scenario where, upon heating and/or annealing for sufficient time, the scale of a microstructure tends to reduce (e.g. grains become smaller, secondary phase particles separate into smaller particles and become more finely dispersed, often reverting to high aspect ratio or high surface-area-to-volume ratio configurations. True or False
Answer:
1) correct answer is ii larger size
2) false, 3) false, 4) true, 5) true, 6) true
Explanation:
In this exercise, the answer is asked if the statement is true.
1) in the rapid cooling speed, there is no thermodynamic equilibrium, so the secondary phases do not have time to transform into the main one, therefore many phases appear in the products,
the correct answer is ii
2) False. The transformation of a material to the glass state requires a fixed temperature and rapid changes to reach this temperature,
3) False. The most stable state is the crystalline state, the glass states are metastable, their Gibbs energy is not the lowest possible and they must transition to the crystalline state over time, it can be years or centuries.
4) True. The melting and freezing temperatures change for each material, within the same material it always has the same value, since it corresponds to a change in the state of the system.
5) true. Cast iron is called gray because of the impurities inside that have not had time to move due to rapid cooling.
6) True. The microstructure is reduced in the process of cooling and heating
arrange 3 identical resistors in all the possible combinations and calculate the equivalent resistance. the resistance for each resistor is 200 ohms
Explanation:
All R's in series: just add them together : 200 + 200 + 200 Ω = 600Ω
One in series with two in parallel :
= 200 Ω + 200*200/(200+200) Ω = 300Ω
All three in parallel :
R = 1 / (1/200 + 1/200 + 1/200) = 66.7 Ω
Derive an expression for ΔS for the reversible. isothermal expansion of one mole of a gas that obeys the van der Waals equation. Use your result to calculate ΔS
for the isothermal compression of ethane from 10.0 L/mol to 1.00 L/mol at 400 K. Compare your result to what you would get using the ideal gas equation.
By Using the van der Waals equation, ΔS is -1496 J/(mol K).
The negative sign indicates that the process is irreversible and that the entropy of the system decreases. Comparing to the ideal gas result, we see that the van der Waals equation predicts a larger decrease in entropy, which is expected since it takes into account the attractive and repulsive forces between gas molecules.
The van der Waals equation for one mole of gas is given by:
\((P + a/v^2) (v - b) = RT\)
where P is the pressure, v is the molar volume, T is the temperature, R is the gas constant, and a and b are constants that depend on the properties of the gas.
For a reversible, isothermal expansion of one mole of gas at constant temperature T, we have:
ΔS \(= q_{rev} / T\)
where ΔS is the change in entropy and \(q_{rev}\) is the heat absorbed by the system in a reversible process. We can find \(q_{rev}\) from the first law of thermodynamics:
\(dU = q_{rev} + PdV\)
where dU is the change in internal energy and PdV is the work done by the system in a reversible expansion. Since the process is reversible, we can use the van der Waals equation to express P as a function of v:
\(P = RT/(v-b) - a/v^2\)
Substituting this into the above equation and integrating from the initial volume V1 to the final volume V2, we get:
ΔU = \(q_{rev} + \int\limits \, V1V2 [RT/(v-b) - a/v^2] dV\)
Solving this integral and rearranging, we get:
\(q_{rev}\) = ΔU - RT ln [(V2-b)/(V1-b)] + a/V2 - a/V1
Substituting this into the equation for ΔS, we get:
ΔS = (ΔU - RT ln [(V2-b)/(V1-b)] + a/V2 - a/V1) / T
For the isothermal compression of ethane from 10.0 L/mol to 1.00 L/mol at 400 K, we can use the van der Waals constants for ethane: a = 5.536 \(L^2 atm/mol^2\) and b = 0.06513 L/mol. Using the ideal gas equation, we would have:
Δ\(S_{ideal}\) = nR ln(V1/V2) = -8.314 J/(mol K) ln(10.0/1.00) = 18.17 J/(mol K)
Using the van der Waals equation, we can calculate ΔS as:
ΔU = \(U2 - U1 = \int\limits \, V1V2 [RT/(v-b) - a/v^2] dV\) = -1418 J/mol
\(q_{rev}\) = ΔU - RT ln [(V2-b)/(V1-b)] + a/V2 - a/V1 = -1418 - 8.314*400 ln(1.00/0.06503) + 5.536/1.00 - 5.536/10.0 = -1496 J/mol
ΔS = \(q_{rev}\) / T = -1496 J/(mol K)
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