GATE Exam > GATE Questions > The standard deviation of any data x_(y) is g...
Start Learning for Free
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)?
Most Upvoted Answer
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).
|
Explore Courses for GATE exam
|
|
Similar GATE Doubts
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)?
Question Description
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)?.
Solutions 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)? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of 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)? defined & explained in the simplest way possible. Besides giving the explanation of
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)?, a detailed solution 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)? has been provided alongside types of 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)? theory, EduRev gives you an
ample number of questions to practice 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)? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup