A chemical reaction has a Q10 of 3. Which of the following rates characterizes this reaction?
a. a rate of 6 at 20°C and 2 at 30°C
b. a rate of 6 at 30°C and 2 at 20°C
c. a rate of 9 at 20°C and 3 at 30°C
d. a rate of 9 at 40°C and 3 at 20°C
e. a rate of 12 at 10°C and 4 at 20°C

Answers

Answer 1

A chemical reaction has a Q10 of 3 option  c. a rate of 9 at 20°C and 3 at 30°C is the rates that characterizes this reaction

The Q10 value is a measure of how much the rate of a chemical reaction changes with a 10°C change in temperature. A Q10 of 3 indicates that the rate of the reaction will increase by a factor of 3 when the temperature is raised by 10°C.

Looking at the answer choices, we can see that option a and b have a Q10 value of 2, which is not the same as the given Q10 value of 3. Option e has a Q10 value of 4, which is also not the same.

Option d has a Q10 value of 3, but the rates given are at 20°C and 40°C, which is not a 10°C change in temperature.

Therefore, the only option that fits the given Q10 value and has rates that are 10°C apart is option c, which has a rate of 9 at 20°C and 3 at 30°C. Therefore, the answer is c.

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Answer 2

Option c states that the rate of the reaction is 9 at 20°C and 3 at 30°C. The ratio of rates between 20°C and 30°C is 9/3 = 3, which matches the Q10 value of 3.  

c. a rate of 9 at 20°C and 3 at 30°C

The Q10 value is a measure of the temperature sensitivity of a reaction, and it is defined as the factor by which the rate of a reaction changes for every 10-degree Celsius change in temperature. A Q10 value of 3 indicates that the rate of the reaction increases by a factor of 3 for every 10-degree Celsius increase in temperature.

This means that the rate of the chemical  reaction is consistent with the temperature sensitivity indicated by the given Q10 value, making option c the correct answer.

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Related Questions

Which part of John Dalton’s theory was disproved by J.J. Thomson’s cathode-ray tube experiments?(1 point)

Atoms can form new substances.


Atoms are indivisible.


Atoms are small.


Atoms can join together.

Answers

atoms are indivisible.

The part of John Dalton’s theory was disproved by J.J. Thomson’s cathode-ray tube experiments is; "Atoms are indivisible".

One of the major postulates of the Dalton's atomic theory is that the atoms of elements are "indivisible".

This idea resounds from the earliest ideas of Greek philosophers concerning the atom.

Through the cathode-ray tube experiments, J.J. Thomson discovered sub-atomic particles.

This nullified the idea that atoms are indivisible.

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a 1.50 g sample of methanol (ch3oh) is placed in an evacuated 1.00 l container at 30. oc. calculate the pressure (in torr) in the container if all of the methanol is vaporized. (assume the ideal gas law, pv

Answers

The pressure in the cοntainer, when all οf the methanοl is vapοrized, is apprοximately 850.44 tοrr.

How tο calculate the pressure in the cοntainer using the ideal gas law?

Tο calculate the pressure in the cοntainer using the ideal gas law, we can use the fοrmula:

PV = nRT

Where:

P is the pressure

V is the vοlume

n is the number οf mοles

R is the ideal gas cοnstant (0.0821 L·atm/(mοl·K) οr 62.36 L·tοrr/(mοl·K))

T is the temperature in Kelvin

First, let's cοnvert the given temperature frοm Celsius tο Kelvin:

T = 30°C + 273.15 = 303.15 K

Next, we need tο determine the number οf mοles οf methanοl. We can use the mοlar mass οf methanοl (CH₃OH) tο calculate this:

Mοlar mass οf CH₃OH = 12.01 g/mοl (C) + 1.01 g/mοl (H) + 16.00 g/mοl (O) + 1.01 g/mοl (H) = 32.04 g/mοl

Number οf mοles = Mass / Mοlar mass

Number οf mοles = 1.50 g / 32.04 g/mοl = 0.0468 mοl

Nοw we can substitute the values intο the ideal gas law equatiοn tο calculate the pressure:

P * 1.00 L = 0.0468 mοl * 0.0821 L·atm/(mοl·K) * 303.15 K

Sοlving fοr P:

P = (0.0468 mοl * 0.0821 L·atm/(mοl·K) * 303.15 K) / 1.00 L

P = 1.119 atm

Finally, we can cοnvert the pressure tο tοrr by using the cοnversiοn factοr:

1 atm = 760 tοrr

P = 1.119 atm * 760 tοrr/atm

P = 850.44 tοrr

Therefοre, the pressure in the cοntainer, when all οf the methanοl is vapοrized, is apprοximately 850.44 tοrr.

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if 669.8 coulombs was used in an electrolytic cell, how many grams of zn would we expect to be plated from a zn2 solution unto an electrode? write your answer to 4 decimals. do not include units.

Answers

If 669.8 coulombs was used in an electrolytic cell, the grams of Zn would we expect to be plated from a Zn²⁺ solution into an electrode is 0.22 g.

The reaction of the reduction of the Zn  as:

Zn²⁺ + 2e⁻ → Zn (s)

Molar mass of Zn = 65.39 g

The equivalent of Zn²⁺ :

The Equivalent = atomic mass/equivalent weight

The Equivalent Zn²⁺ = 65.39/32.69

The Equivalent Zn²⁺ = 2 mol

1 Faraday = 96500 coulomb

2 F = 96500 * 2 coulomb

2 F = 1,93,000 coulomb

Charge = moles × 1,93,000

moles = 0.0034 mol

The moles = mass / molar mass

The mass = moles × molar mass

The mas = 0.0034 / 65.39

The mass = 0.22 g

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Hey can somebody please give me the answer to this.

Hey can somebody please give me the answer to this.

Answers

Answer:

it is d or b hope this help

Answer:

A

Explanation:

How many grams of liquid water are produced when 60 grams of ice melt?

Answers

Answer: 60 Grams of water are produced, because you can't get more than the amount that was in the original ice cube

Explanation:

You were given a 100. G wine sample to verify its age. Using tritium dating you observe that the sample has 0. 688 decay events per minute. Tritium has a half life of 12. 3 and fresh water exhibits 5. 5 decay events per minute per 100g. What year was the wine produced?.

Answers

Wine was produced 37 years ago (1984 as usual year 15,2021) that is shown in the calculations below.

Reaction rate is calculated using the formula rate = Δ[C]/Δt, where Δ[C] is the change in product concentration during time period Δt. The rate of reaction can be observed by watching the disappearance of a reactant or the appearance of a product over time.

The time can be represented as follows:

t= 2.303\∧ log A0/A

∧= 0.693/t 1/2

The rate of a reaction is proportional to the reciprocal of the time taken. Rate α 1 time Rate is inversely proportional to time. Units: s-1, min-1 etc.

The given parameters are as follows:

t1/2=12.3

A0=5.5

A=0.688

t= 2.303/(0.693/12.3) log (5.5/0.688)

t=36.9

t=37 years

Thus, wine was produced 37 years ago (1984 as usual year 15,2021)

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Any 1 help with Chemistry here??

Answers

Nope sorry bro:((((((

11. Determine the number of molecules present in 0.25 mol of N204

Answers

The number of molecules : N = 1.505 x 10²³

Further explanation

Given

0.25 mol N₂O₄

Required

The number of molecules

Solution

Can be formulated

N=n x No

N = number of particles

n = mol

No = Avogadro's = 6.02.10²³

N = 0.25 x 6.02.10²³

N = 1.505 x 10²³

Calculate the mass of ammonium sulfide (NH4)2S in 3.00 L of a 0.0200 M solution.

Answers

The mass of (NH4) 2S in the solution is : Mass = 0.0600 mol × 60.08 g/mol = 3.60 g.

The given molarity and volume of the solution can be used to calculate the number of moles of ammonium sulfide (NH4)2S.Then, the number of moles can be converted to mass using the molar mass of (NH4)2S.Mass of ammonium sulfide (NH4)2S in 3.00 L of a 0.0200 M solution is given by : Mass = moles × molar mass.The number of moles of (NH4)2S can be found using the equation:Molarity = Number of moles / Volume.Rearranging this equation, we get:Number of moles = Molarity × Volume Number of moles of (NH4)2S = 0.0200 M × 3.00 L.Number of moles of (NH4)2S = 0.0600 mol.The molar mass of (NH4)2S can be calculated by summing the molar masses of ammonium (NH4) and sulfide (S) ions.Molar mass of (NH4)2S = (2 × Molar mass of NH4) + Molar mass of S= (2 × 14.01 g/mol) + 32.06 g/mol= 60.08 g/mol.

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A chemical bond formed when two atoms share two pairs of electrons is a ________ bond; it is best described as ________.

Answers

Answer:

A chemical bond formed when two atoms share two pairs of electrons is a double bond; it is best described as a covalent bond.

Explanation:

A chemical bond in which two pairs of electrons are shared has to be defined as covalent since ionic bonds don't involve electron sharing. They consist only of electrostatic attraction between ions.

Based on the graph above which statement best describes the relationship between pressure and volume

Based on the graph above which statement best describes the relationship between pressure and volume

Answers

The relationship between volume is inversely proportional. This means that when there are an increase in pressure of molecules that are moving, pushes the molecules closer together, reducing the volume. If the pressure is decreased, the gases are free to move about in a larger volume.

You can see in the graph that the behaviour is: the volume is increasing, so this causes that the pressure increase. The answer would be the third one: pressure increases

limitations of paper chromatography​

Answers

Answer:

Limitations of Paper Chromatography.

1.Large quantity of sample cannot be applied on paper chromatography.

2.In quantitative analysis paper chromatography is not effective.

3.Complex mixture cannot be separated by paper chromatography.

4.Less Accurate compared to HPLC or HPTLC.

Answer:

Limitations of paper chromatography​ is described below in complete details.

Explanation:

Limitations of Paper Chromatography

A huge number of representations cannot be employed in paper chromatography.In a quantitative investigation, paper chromatography is not sufficient.The complex compound cannot be separated by paper chromatography.Less Accurate correlated to HPLC or HPLC.

If you ever felt dizzy after Spinning around and around on an amusement park ride, which of the following was to blame.

A stirrup
B ossicles
C Tympanic membrane
D Semicircular canal
E The chili cheese fries right before the ride

Answers

Answer:

C Tympanic membrane

Explanation:

due to the nerved gathered inside the membrane can cause a nauseous feeling.

How were clouds made?

Answers

Answer: Clouds form when the invisible water vapor in the air condenses into visible water droplets or ice crystals.

Explanation:

Clouds form when the invisible water vapor in the air condenses into visible water droplets or ice crystals. For this to happen, the parcel of air must be saturated, i.e. unable to hold all the water it contains in vapor form, so it starts to condense into a liquid or solid
form.

6. A piece of solid gold was heated from 274K to 314K. 35.73 of energy was needed to raise the temperature.
What mass of gold was present?

Answers

From Q = mcΔT, we can rearrange the equation to solve for mass, m = Q/cΔT. The specific heat capacity, c, of solid gold is 0.129 J/g °C. I'm assuming that the energy is given in joules, as it's not specified in the question as written.

m = Q/cΔT = (35.73 J)/(0.129 J/g °C)(40.85 °C - 0.85°C)

m = 6.92 g of gold was present  

The heat of the substance depends on the mass, specific heat, and temperature change of the system. The mass of gold that was present was, 6.92 g.

What is mass?

Mass of the substance in the thermodynamic system is the ratio of the heat to the product of the specific heat and the temperature change. It is given as,

\(\rm m = \rm \dfrac{Q}{c \Delta T}\)

Given,

Heat (Q) = 35.73 J

Specific heat (c) = 0.129 J/g °C

Temperature change = 40 degrees celsius

Mass of gold is calculated as:

35.73 ÷ (0.129) (40) = 6.92 gm

Therefore, 6.92 gms of gold is present.

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What is forensic
science?

Answers

Answer:

"Forensic science is a critical element of the criminal justice system. Forensic scientists examine and analyze evidence from crime scenes and elsewhere to develop objective findings that can assist in the investigation and prosecution of perpetrators of crime or absolve an innocent person from suspicion."

Explanation:

Strictly from: https://www.justice.gov/olp/forensic-science#:~:text=Forensic%20science%20is%20a%20critical,an%20innocent%20person%20from%20suspicion.

Forensic science is any science used to aid in a criminal/legal investigation

you have 5L of a stock solution of 6M AgNO3, you need a 3M AgNO3 strength solution for a lab assignment. What volume of 3M could you make from the stock solution.

Answers

The volume of 3M from making stock solution is 7.5 liters.

To make a 3M solution from a 6M stock solution of AgNO3, we need to add water to the stock solution. Let us assume x liters of 3M solution from 5 liters of the 6M stock solution.

The amount of AgNO3 in the 5 liters of the stock solution is:

5 L * 6 mol/L = 30 mol AgNO3

To make a 3M solution, we need:

3 mol/L * x L = 3x mol AgNO3

We can use the formula:

C1V1 = C2V2

where,

C1 = concentration of the stock solution

C2 = concentration of the final solution

V1 = volume of stock solution

V2 = volume of the final solution

In this case, we know that:

C1 = 6 M

C2 = 3 M

V1 = x

V2 = 5 L + x

In this case, we know that:

C1 = 6 M

C2 = 3 M

V1 = x

V2 = 5 L + x

let us substitute the above values in the formula

6 M * V1 = 3 M * (5 L + x)

Solving for x, we get:

x = 2.5 L

Therefore, to make a 3M solution from a 6M stock solution, a final volume of 5 + 2.5 = 7.5 liters to get a 3M solution.

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DRAW THE STRUCTURE (ASAPP)

DRAW THE STRUCTURE (ASAPP)

Answers

The structure of 3-ethyl-2 fluroheptane is attached. The chemical formula of the compound is C9H19F.

What is 3-ethyl-2 fluroheptane?

It is an organic compound which has one pair of hydrogen and acceptor and zero hydrogen bond donor.

The chemical formula of the compound is \(C_9H_1_9_F.\)

Thus, the structure of 3-ethyl-2 fluroheptane is attached. The chemical formula of the compound is C9H19F.

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DRAW THE STRUCTURE (ASAPP)

aluminum oxide is an ionic substance with formula al2o3. explain what this formula means.

Answers

Answer:

the 2 means there are 2 al and the 3 means there are 3 o

Explanation:

I hope this helps

In the chemical reaction, Fe(s) + 2HCl(aq) →FeCl2 (aq) + gas, the gas evolved is A helium. B oxygen.C chlorine.D hydrogen. ​

Answers

Answer:

The correct option is A Hydrogen

Explanation:

The correct option is A Hydrogen

Iron on reaction with hydrochloric acid, replaces the hydrogen from the acid and forms ferrous chloride. Hydrogen gas is evolved during the reaction in the form of bubbles.

Fe (s) + 2HCl (aq) → FeCl2 (aq) +H2 (g)

which ideas john dalton have

Answers

Answer:

He is best known for introducing the atomic theory into chemistry, and for his research into colour blindness, sometimes referred to as Daltonism in his honour.

Explanation:

Answer:

Law of conservation of mass and the law of definite proportions could be explained using the idea of atoms.

PLS HELP!!!!!! I will give brainly-ist
I am doing a science lab report on edge.
Also it is on comparing different material on earths surface absorb heat and to understand how to angle of insolation affects the heating of earths surface.
I have two questions.
1. what is your hypothesis for this experiment?
2. what methods are you using to test this hypothesis?
BTW this is science

Answers

Answer:

A biology investigation usually starts with an observation—that is, something that catches the biologist’s attention. For instance, a cancer biologist might notice that a certain kind of cancer can't be treated with chemotherapy and wonder why this is the case. A marine ecologist, seeing that the coral reefs of her field sites are bleaching—turning white—might set out to understand why.

How do biologists follow up on these observations? How can you follow up on your own observations of the natural world? In this article, we’ll walk through the scientific method, a logical problem-solving approach used by biologists and many other scientists.

Explanation:

how does kinetic molecular theory explain gas pressure inside a balloon? select the correct answer below: gas particles are large and only a few fit inside a balloon, inflating the walls. the atmospheric pressure is greater than the pressure of the gas inside the balloon, so the balloon expands gas particles are very small, and many will fill the inside of the balloon until they are all stationary and touching each other. the pressure exerted by a gas in the balloon results from collisions between the gas molecules and the balloon walls.

Answers

Answer:

Therefore the answer is d) The pressure exerted by a gas in the balloon results from collisions between the gas molecules and the balloon walls.

Explanation:

You are given a piece of paper and a match. The paper has a mass of 2.5 g. You then light the match and light the piece of paper on fire. After it burns, the remaining bits of paper weigh 0.5 g. Does this demonstration violate the conservation of mass? Explain why or why not?

Answers

Answer:

No

Explanation:

No. The demonstration does not violate the conservation of mass.

The law of conservation of mass states that mass can neither be created nor destroyed in a reaction. However, mass can be converted from one form to another during the reaction.

In this case, even though the remaining bits of paper weigh 0.5 g while the original paper weighed 2.5 g, the ashes and smoke/gas from the burning will all add up to the lost weight of the paper.

The burned part has been converted into other forms. If the smoke/gas and the ashes are properly captured, they will mark up with the weight of the remaining paper to give the weight of the original paper.

The answer to the question on whether the burning violates the law of conservation of mass, we say that;

No, the demonstration does not violate the conservation of mass because the mass formed still adds up to the original mass.

The law of conservation of mass states that mass in an isolated system, matter can neither be created nor destroyed by chemical reactions or physical transformations. However, it could involve change in mass from one form to another.  

Now, we are told that the remaining bits of paper has a mass of 0.5 g while the original paper has a mass 2.5 g. The reason why the mass left is 0.5 kg is because the ashes and smoke from the burning will add up to give the original weight of the paper.

Thus, since when the ashes, and smoke add up to still give the original mass of 3.5 g, it means that the mass is neither created nor destroyed

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You kick a 1.2 kg soccer ball at 24 m/s what is the mechanical kinetic energy of the soccer ball

Answers

Answer:

E = 345.6  J

Explanation:

Given that,

Mass of a soccer ball, m = 1.2 kg

Speed of the soccer ball, v = 24 m/s

We need to find the mechanical kinetic energy of the soccer ball. The formula for the kinetic energy of an object is given by :

\(E=\dfrac{1}{2}mv^2\)

Put all the values in the above formula

\(E=\dfrac{1}{2}\times 1.2\times (24)^2\\\\E=345.6\ J\)

So, the kinetic energy of the soccer ball is 345.6  J.

Helppp me plzzzzz!!!

Helppp me plzzzzz!!!

Answers

Answer:

they use echos to communicate

Explanation: dolphins make echos in order to signal there partners, and others in there pod. or they can use it to signal for prey, or signal that they are in trouble.

Answer: A good example for this one is bats.

Explanation: While a couple of bats are known to emanate the beats through their nose, most bats will in general express them. The recurrence is estimated to be around 200 heartbeats each second, nonetheless, the time frame can fluctuate with the types of a bat. Credits: Adi Ciurea/Shutterstock A bat's squeaks are more rewarding than Ben's on the grounds that with higher recurrence comes higher-goal or more distinctive detail – they are equipped for evading deterrents no more extensive than a human hair! They use it to find and explore towards prey, for example, a hurrying bug.   As the distinguished sound waves that structure the pictures are basically echoes, they should adjust to a standard — the recovered train of heartbeats should be sufficiently noisy to get back to the sender, yet short sufficient with the end goal that the reverberation of the sender returns before the following one withdraw.

what is the total volume of solution for each of the dilutions of albumin stock solution in the case study?

Answers

The total volumes of solution for the 1:5, 1:10, 1:20, and 1:40 dilutions are 10.4 mL, 11 mL, 14 mL, and 19 mL, respectively.

In the case study, four dilutions of an albumin stock solution were prepared: 1:5, 1:10, 1:20, and 1:40. The total volume of solution for each dilution can be calculated using the formula:

\( \text{Total volume} = \frac{\text{Volume of stock solution}}{\text{Dilution factor}} \)

For example, if 2 mL of the stock solution was diluted to 10 mL to create a 1:5 dilution, the calculation would be:

\( \text{Total volume} = \frac{2 \, \text{mL}}{5} = 0.4 \, \text{mL} \)

Thus, the total volume of solution for the 1:5 dilution is 10 mL + 0.4 mL = 10.4 mL.

Similarly, the total volumes for the other dilutions can be calculated:

1:10 dilution: total volume = 10 mL + 1 mL = 11 mL

1:20 dilution: total volume = 10 mL + 4 mL = 14 mL

1:40 dilution: total volume = 10 mL + 9 mL = 19 mL

The total volumes of solution for the 1:5, 1:10, 1:20, and 1:40 dilutions are 10.4 mL, 11 mL, 14 mL, and 19 mL, respectively.

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which list consists of elements that have the most similar chemical properties? group of answer choices cu, zn, fe k, ca, br cs, na, k mg, al, si

Answers

Magnesium and Lithium have similar chemical properties.

Chemical characteristics of elements in the same group on the periodic table are comparable. This is because the number of electrons in each atoms' highest-occupied energy level is the same.

Since both Li lithium and K potassium are members of the first group of alkali metals, they share a number of chemical characteristics. Some pairs of diagonally adjacent items in the second and third periods have a diagonal relationship. The first 20 elements in the periodic table are listed here.

Examples of pairings that share characteristics include lithium (Li) and magnesium (Mg), beryllium (Be) and aluminum (Al), and boron (B) and silicon (Si). Both silicon and boron are semiconductors.

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A large fish tank is initially filled with 30 litres of fresh water. You begin to fill the tank by slowly pouring in water with salt concentration of 35 grams per litre (approximate salinity of sea water) at a rate of 2 litres per minute. At the same time, the (perfectly mixed) fluid in the tank is drained from the bottom at a rate of 1 litre per minute. 1. Determine the volume of water in the tank at time t. [1 mark] 2. Let S(t) denote the amount of salt in the fish tank at time t in grams. Show that S(t) satisfies the ODE S

(t)=70−
t+30
S

. Write down the appropriate initial condition for the ODE as well. [2 marks] 3. What order is this ODE? Is it linear? Is it separable? [1 mark] 4. Solve the initial value problem to find S(t) using the method of integrating factors. [3 marks] 5. What is the salt concentration in the tank as t→[infinity] ? [1 mark] Part B: Double tanks Next you hook up two fish tanks in a loop so that there is a pipe from tank A to tank B, and also a pipe from tank B back to tank A. Two pumps are added so that you can control the flow rate in each pipe. Initially tank A contains 80 litre of fresh water and tank B 60 litres of fresh water. You begin to pour salt water with concentration 35 grams per litre into tank A at a rate of 2 litres per minute. To keep the tanks from overflowing, you set your pumps so that water is flowing at a constant rate of 4 litres per minute from tank A to tank B, and 2 litre per minute from tank B to tank A. You also put a drain in tank B so that fluid is draining at a rate of 2 litres per minute. 1. Sketch a diagram of the tank setup with arrows for flows entering and leaving each tank. [
1 mark]

2. Let P(t) and Q(t) denote the amount of salt in tank A and tank B respectively. Show that P and Q satisfy a system of ODE's in the form of
P

(t)
Q

(t)


=c
1

P(t)+c
2

Q(t)+c
3


=c
4

P(t)+c
5

Q(t)

where c
1

,c
2

,c
3

,c
4

and c
5

are constants. Determine the constant c
1

,c
2

,c
3

,c
4

,c
5

and write down appropriate initial conditions. [2 marks] 3. Show that the system of ODE's can be converted into the following second order ODE for P(t) P
′′
(t)=−
60
7

P

(t)−
600
1

P(t)+
3
14

State the initial conditions for this ODE. [2 marks] 4. Solve this second order ODE to find P(t), and hence Q(t) as well.

Answers

1. The volume of water in the tank at time t is given by the equation

Volume(t) = 30 + t.

2.The appropriate initial condition for the ODE is S(0) = 0, as there is no salt initially in the tank.

3. It is not separable because the variables S(t) and t are not separable on opposite sides of the equation.

4. The solution can be expressed in terms of the integral as:
\(S(t) = (70 * \int e^{(t^{2/2} + 30t)} dt) / e^{(t^{2/2} + 30t)})\)

5.  the salt concentration in the tank as t→infinity is zero.

1. To determine the volume of water in the tank at time t, we need to consider the rate at which water is being added and drained. The tank is being filled at a rate of 2 liters per minute and drained at a rate of 1 liter per minute.

Since the tank starts with an initial volume of 30 liters, the volume of water in the tank at time t can be calculated using the equation:
Volume(t) = Initial volume + (Rate of filling - Rate of draining) * t

Volume(t) = 30 + (2 - 1) * t

So, the volume of water in the tank at time t is given by the equation

Volume(t) = 30 + t.

2. Let S(t) denote the amount of salt in the fish tank at time t in grams.

To show that S(t) satisfies the ODE S'(t) = 70 - (t+30)S(t),

we need to take the derivative of S(t) with respect to t and substitute it into the given ODE.

Taking the derivative of S(t), we have:

S'(t) = 0 - (1+0)S(t) + 0

S'(t) = -S(t)

Substituting this into the given ODE, we get:

-S(t) = 70 - (t+30)S(t)

Simplifying the equation, we have:

S'(t) = 70 - (t+30)S(t)

Therefore, S(t) satisfies the ODE S'(t) = 70 - (t+30)S(t).

The appropriate initial condition for the ODE is S(0) = 0,

as there is no salt initially in the tank.

3. This ODE is a first-order linear ordinary differential equation. It is not separable because the variables S(t) and t are not separable on opposite sides of the equation.

4. To solve the initial value problem for S(t) using the method of integrating factors, we first rewrite the ODE in standard form:

S'(t) + (t+30)S(t) = 70

The integrating factor is given by:
\(\mu(t) = e^{(\int (t+30) dt)} = e^{(t^2/2 + 30t)\)

Multiplying both sides of the equation by μ(t), we have:
\(e^{(t^2/2 + 30t)} * S'(t) + e^{(t^2/2 + 30t)} * (t+30)S(t) = 70 * e^{(t^2/2 + 30t)\)

Applying the product rule to the left side of the equation, we get:
\((e^{(t^{2/2} + 30t) * S(t))' = 70 * e^{(t^{2/2} + 30t)})\)

Integrating both sides of the equation with respect to t, we have:
\(\int (e^{(t^2/2 + 30t)} * S(t))' dt = \int (70 * e^{(t^2/2 + 30t))} dt\)

Using the fundamental theorem of calculus, the left side becomes:
\(e^{(t^2/2 + 30t)} * S(t) = \int (70 * e^{(t^2/2 + 30t))} dt\)

Simplifying the right side by integrating, we get:
\(e^{(t^2/2 + 30t)} * S(t) = 70 * \int e^{(t^2/2 + 30t)} dt\)

At this point, the integration of \(e^{(t^2/2 + 30t)\) becomes difficult to express in terms of elementary functions.

Hence, the solution can be expressed in terms of the integral as:
\(S(t) = (70 * \int e^{(t^2/2 + 30t)} dt) / e^{(t^2/2 + 30t)\)

5. As t approaches infinity, the exponential term \(e^{(t^2/2 + 30t)\) becomes very large, causing the salt concentration S(t) to approach zero. Therefore, the salt concentration in the tank as t→infinity is zero.

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The salt concentration in the tank as t approaches infinity is 70/3.

1. To determine the volume of water in the tank at time t, we need to consider the rate at which water is being poured into the tank and the rate at which water is being drained from the bottom.

At a rate of 2 litres per minute, water is being poured into the tank. So after t minutes, the amount of water poured into the tank is 2t litres.

At a rate of 1 litre per minute, water is being drained from the tank. So after t minutes, the amount of water drained from the tank is t litres.

Since the tank was initially filled with 30 litres of fresh water, the volume of water in the tank at time t is given by:
Volume(t) = 30 + 2t - t
Volume(t) = 30 + t

2. Let S(t) denote the amount of salt in the fish tank at time t. To determine the ODE for S(t), we need to consider the salt being poured into the tank and the salt being drained from the tank.

The salt concentration in the water being poured into the tank is 35 grams per litre. So the amount of salt being poured into the tank per minute is 35 * 2 = 70 grams.

The amount of salt being drained from the tank per minute is S(t)/Volume(t) * 1.

Therefore, the ODE for S(t) is:
S'(t) = 70 - S(t)/Volume(t)

The initial condition for this ODE is S(0) = 0, since there was no salt in the tank initially.

3. The ODE S'(t) = 70 - S(t)/Volume(t) is a first-order linear ODE. It is not separable since the variables S(t) and Volume(t) are mixed together.

4. To solve the initial value problem for S(t), we can rewrite the ODE as:
Volume(t) * S'(t) + S(t) = 70 * Volume(t)

This is a linear ODE of the form y'(t) + p(t)y(t) = g(t), where p(t) = 1/Volume(t) and g(t) = 70 * Volume(t).

To solve this type of ODE, we can multiply both sides by an integrating factor, which is the exponential of the integral of p(t).

The integrating factor is exp(integral of 1/Volume(t) dt) = exp(ln(Volume(t))) = Volume(t).

Multiplying both sides of the ODE by the integrating factor, we get:
Volume(t) * S'(t) + S(t) = 70 * Volume(t)
Volume(t) * S'(t) + Volume(t) * S(t) = 70 * Volume(t)^2
( Volume(t) * S(t) )' = 70 * Volume(t)^2

Integrating both sides with respect to t, we get:
Volume(t) * S(t) = 70/3 * Volume(t)^3 + C

S(t) = 70/3 * Volume(t)^2 + C/Volume(t)

Using the initial condition S(0) = 0, we can solve for C:
0 = 70/3 * 30^2 + C/30
C = -70000

Therefore, the solution for S(t) is:
S(t) = 70/3 * Volume(t)^2 - 70000/Volume(t)

5. As t approaches infinity, the volume of water in the tank becomes very large. In this case, we can approximate the volume of the tank as t, since the rate at which water is being poured in is 2 litres per minute. So the salt concentration in the tank as t approaches infinity is given by:
S(t)/Volume(t) = (70/3 * t^2 - 70000/t) / t
As t approaches infinity, the second term (-70000/t) approaches 0, so the salt concentration in the tank as t approaches infinity is:
S(t)/Volume(t) = 70/3 * t^2 / t = 70/3 * t

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Let c(x,y,z,t) denote the concentration of a pollutant (the amount per unit volume). (a) What is an expression for the total amount of pollutant in the region R ? (b) Suppose that the flow J of the pollutant is proportional to the gradient of the concentration. (Is this reasonable?) Express conservation of the pollutant. (c) Derive the partial differential equation governing the diffusion of the pollutant.

Answers

∂c/∂t represents the rate of change of concentration with respect to time, D is the diffusion coefficient, and ∇·(D∇c) represents the divergence of the diffusion flux.

(a) The total amount of pollutant in the region R can be found by integrating the concentration c(x,y,z,t) over the volume V of the region R. Mathematically, it can be expressed as:

Total amount of pollutant = ∫∫∫V c(x,y,z,t) dV

(b) It is reasonable to assume that the flow J of the pollutant is proportional to the gradient of the concentration, as this relationship is based on Fick's law of diffusion. According to Fick's law, the flow of a substance is proportional to the concentration gradient. Mathematically, it can be expressed as:

J = -D ∇c

Where J is the flow of the pollutant, D is the diffusion coefficient, and ∇c is the gradient of the concentration.

(c) To derive the partial differential equation governing the diffusion of the pollutant, we can apply the continuity equation, which states that the rate of change of the concentration in a given volume is equal to the divergence of the flow. Mathematically, it can be expressed as:

∂c/∂t = -∇·J

Using the relationship from part (b), we can substitute it into the continuity equation:

∂c/∂t = ∇·(D∇c)

This is the partial differential equation governing the diffusion of the pollutant, where ∂c/∂t represents the rate of change of concentration with respect to time, D is the diffusion coefficient, and ∇·(D∇c) represents the divergence of the diffusion flux.

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