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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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