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The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)?
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The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(...
Standard Deviation of Maxwell's Speed
The standard deviation of Maxwell's speed, denoted as Δv, represents the spread or dispersion of the speeds of molecules in an assembly. It is a measure of the average deviation of the speeds from the mean speed. The standard deviation can be calculated using the formula:
Δv = sqrt(⟨v^2⟩ - ⟨v⟩^2)
where ⟨v^2⟩ is the average of the square of the speeds and ⟨v⟩ is the average speed.
Maxwell-Boltzmann Distribution
The distribution of speeds of molecules in a gas at a given temperature T is described by the Maxwell-Boltzmann distribution. According to this distribution, the probability of finding a molecule with speed v is given by:
P(v) = (4πv^2) * (M/(2πRT))^3/2 * exp(-Mv^2/(2RT))
where M is the molar mass of the molecules, R is the gas constant, and T is the temperature.
Calculating the Standard Deviation
To calculate the standard deviation of Maxwell's speed, we need to find ⟨v^2⟩ and ⟨v⟩.
Finding ⟨v^2⟩
⟨v^2⟩ represents the average of the square of the speeds. We can calculate it by integrating v^2 * P(v) over all possible speeds:
⟨v^2⟩ = ∫(v^2 * P(v)) dv
Substituting the expression for P(v) and performing the integration, we find:
⟨v^2⟩ = (3RT)/M
Finding ⟨v⟩
⟨v⟩ represents the average speed of the molecules. We can calculate it by integrating v * P(v) over all possible speeds:
⟨v⟩ = ∫(v * P(v)) dv
Substituting the expression for P(v) and performing the integration, we find:
⟨v⟩ = sqrt((8RT)/(πM))
Calculating the Standard Deviation
Now that we have ⟨v^2⟩ and ⟨v⟩, we can calculate the standard deviation Δv using the formula:
Δv = sqrt(⟨v^2⟩ - ⟨v⟩^2)
Substituting the values we found earlier:
Δv = sqrt((3RT)/M - (8RT)/(πM))
Simplifying the expression:
Δv = sqrt((3π-8)/(πM)) * sqrt(RT)
Δv = 0.7 * sqrt(RT/M)
Therefore, the standard deviation of Maxwell's speed is approximately 0.7 * sqrt(RT/M). Hence, the correct option is (a) 0.7 * sqrt(RT/M).
The standard deviation of Maxwell's speed, denoted as Δv, represents the spread or dispersion of the speeds of molecules in an assembly. It is a measure of the average deviation of the speeds from the mean speed. The standard deviation can be calculated using the formula:
Δv = sqrt(⟨v^2⟩ - ⟨v⟩^2)
where ⟨v^2⟩ is the average of the square of the speeds and ⟨v⟩ is the average speed.
Maxwell-Boltzmann Distribution
The distribution of speeds of molecules in a gas at a given temperature T is described by the Maxwell-Boltzmann distribution. According to this distribution, the probability of finding a molecule with speed v is given by:
P(v) = (4πv^2) * (M/(2πRT))^3/2 * exp(-Mv^2/(2RT))
where M is the molar mass of the molecules, R is the gas constant, and T is the temperature.
Calculating the Standard Deviation
To calculate the standard deviation of Maxwell's speed, we need to find ⟨v^2⟩ and ⟨v⟩.
Finding ⟨v^2⟩
⟨v^2⟩ represents the average of the square of the speeds. We can calculate it by integrating v^2 * P(v) over all possible speeds:
⟨v^2⟩ = ∫(v^2 * P(v)) dv
Substituting the expression for P(v) and performing the integration, we find:
⟨v^2⟩ = (3RT)/M
Finding ⟨v⟩
⟨v⟩ represents the average speed of the molecules. We can calculate it by integrating v * P(v) over all possible speeds:
⟨v⟩ = ∫(v * P(v)) dv
Substituting the expression for P(v) and performing the integration, we find:
⟨v⟩ = sqrt((8RT)/(πM))
Calculating the Standard Deviation
Now that we have ⟨v^2⟩ and ⟨v⟩, we can calculate the standard deviation Δv using the formula:
Δv = sqrt(⟨v^2⟩ - ⟨v⟩^2)
Substituting the values we found earlier:
Δv = sqrt((3RT)/M - (8RT)/(πM))
Simplifying the expression:
Δv = sqrt((3π-8)/(πM)) * sqrt(RT)
Δv = 0.7 * sqrt(RT/M)
Therefore, the standard deviation of Maxwell's speed is approximately 0.7 * sqrt(RT/M). Hence, the correct option is (a) 0.7 * sqrt(RT/M).
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The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)?
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The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)?.
The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The standard deviation of any data x_(y) is given by Delta x=sqrt((x^(2)))-(x^(2)). For an assembly of molecules of molar mass M at temperature T the standard deviation ofMaxwell's speed is approximately. (a) 0.7sqrt(RT/M) (b) 1.4sqrt(RT/M) (c) 0.7sqrt(M/RT)) (d) 1.4sqrt(M/RT)?.
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