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A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)
Correct answer is '25'. Can you explain this answer?
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A sample contains 10-2 kg each of two substances A and B with half liv...
Given:
- Mass of substance A = Mass of substance B = 10^-2 kg
- Half-life of substance A = 4 s
- Half-life of substance B = 8 s
- Ratio of atomic weights of A and B = 1:2
To find:
- Ratio of amounts of A and B after 16 s
Solution:
1. Calculate the decay constant (λ) for substances A and B using the formula:
λ = ln2 / t1/2
where t1/2 is the half-life of the substance.
For substance A:
λA = ln2 / 4 = 0.1733 s^-1
For substance B:
λB = ln2 / 8 = 0.0866 s^-1
2. Calculate the number of moles of each substance:
nA = m / M, where m is the mass and M is the molar mass of the substance.
Since the ratio of atomic weights of A and B is 1:2, their molar masses are also in the ratio of 1:2. Let the molar mass of A be x, then the molar mass of B is 2x.
For substance A:
nA = 10^-2 / x
For substance B:
nB = 10^-2 / (2x)
3. Calculate the amount of each substance at time t using the formula:
N(t) = N0 e^(-λt)
where N0 is the initial number of atoms/molecules.
For substance A:
NA(t) = NA(0) e^(-λA t)
where NA(0) is the initial number of atoms of A.
For substance B:
NB(t) = NB(0) e^(-λB t)
where NB(0) is the initial number of atoms of B.
4. Calculate the ratio of amounts of A and B at time t:
NA(t) / NB(t) = (NA(0) / NB(0)) (e^(λB - λA)t)
Since NA(0) = NB(0) (since both substances have the same mass), we can simplify the above equation as:
NA(t) / NB(t) = e^(λB - λA)t
Substituting the values of λA, λB and t=16 s, we get:
NA(16) / NB(16) = e^(-0.0866*16 + 0.1733*16) = e^1.3867
5. Convert the ratio of amounts to a percentage:
x/100 = (NA(16) / (NA(16) + NB(16))) * 100
Substituting the value of NA(16) / NB(16) from step 4, we get:
x/100 = (1 / (1 + e^(-1.3867))) * 100
Simplifying, we get:
x/100 = 25 (approx)
Therefore, the required value of x is 25.
- Mass of substance A = Mass of substance B = 10^-2 kg
- Half-life of substance A = 4 s
- Half-life of substance B = 8 s
- Ratio of atomic weights of A and B = 1:2
To find:
- Ratio of amounts of A and B after 16 s
Solution:
1. Calculate the decay constant (λ) for substances A and B using the formula:
λ = ln2 / t1/2
where t1/2 is the half-life of the substance.
For substance A:
λA = ln2 / 4 = 0.1733 s^-1
For substance B:
λB = ln2 / 8 = 0.0866 s^-1
2. Calculate the number of moles of each substance:
nA = m / M, where m is the mass and M is the molar mass of the substance.
Since the ratio of atomic weights of A and B is 1:2, their molar masses are also in the ratio of 1:2. Let the molar mass of A be x, then the molar mass of B is 2x.
For substance A:
nA = 10^-2 / x
For substance B:
nB = 10^-2 / (2x)
3. Calculate the amount of each substance at time t using the formula:
N(t) = N0 e^(-λt)
where N0 is the initial number of atoms/molecules.
For substance A:
NA(t) = NA(0) e^(-λA t)
where NA(0) is the initial number of atoms of A.
For substance B:
NB(t) = NB(0) e^(-λB t)
where NB(0) is the initial number of atoms of B.
4. Calculate the ratio of amounts of A and B at time t:
NA(t) / NB(t) = (NA(0) / NB(0)) (e^(λB - λA)t)
Since NA(0) = NB(0) (since both substances have the same mass), we can simplify the above equation as:
NA(t) / NB(t) = e^(λB - λA)t
Substituting the values of λA, λB and t=16 s, we get:
NA(16) / NB(16) = e^(-0.0866*16 + 0.1733*16) = e^1.3867
5. Convert the ratio of amounts to a percentage:
x/100 = (NA(16) / (NA(16) + NB(16))) * 100
Substituting the value of NA(16) / NB(16) from step 4, we get:
x/100 = (1 / (1 + e^(-1.3867))) * 100
Simplifying, we get:
x/100 = 25 (approx)
Therefore, the required value of x is 25.
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Community Answer
A sample contains 10-2 kg each of two substances A and B with half liv...
∴ Mass ratio of A and B,
∴ x = 25
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A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer?
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A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer?.
A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sample contains 10-2 kg each of two substances A and B with half lives 4 s and 8 s, respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ________. (In integers)Correct answer is '25'. Can you explain this answer?.
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