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Pressure exerted by an ideal gas molecule is given by the expression
- a)P=MC2/V
- b)P=M2/CV2
- c)P=M/VC
- d)P=MC2/3V
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Pressure exerted by an ideal gas molecule is given by the expressiona)...
D is the derived expression.
Most Upvoted Answer
Pressure exerted by an ideal gas molecule is given by the expressiona)...
The expression for pressure exerted by an ideal gas molecule is given by the ideal gas law equation:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
To get the expression for pressure exerted by a single gas molecule, we need to divide both sides by the number of molecules (N) and express the number of moles (n) in terms of the number of molecules (N) using Avogadro's number (NA):
PV/N = (n/NA)RT
PV/N = kT
where k is the Boltzmann constant (R/NA) and T is the temperature in Kelvin.
Now, we can express the volume (V) in terms of the number of molecules (N) using the density (ρ) of the gas:
V = N/ρ
Substituting this into the above equation, we get:
P = kTρ
Using the ideal gas law equation, we can express the density (ρ) in terms of the molar mass (M) and the volume (V):
ρ = M/V
Substituting this into the above equation, we get:
P = kTM/V
Finally, we can express the Boltzmann constant (k) in terms of the gas constant (R) and Avogadro's number (NA):
k = R/NA
Substituting this into the above equation, we get:
P = (R/NA)TM/V
Simplifying this expression, we get:
P = (M/3)V2C2
where C is the root-mean-square speed of the gas molecules, given by:
C = √(3RT/M)
Substituting this into the above equation, we get the final expression for pressure exerted by an ideal gas molecule:
P = (M/3V)C2
Therefore, the correct answer is option 'D'.
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
To get the expression for pressure exerted by a single gas molecule, we need to divide both sides by the number of molecules (N) and express the number of moles (n) in terms of the number of molecules (N) using Avogadro's number (NA):
PV/N = (n/NA)RT
PV/N = kT
where k is the Boltzmann constant (R/NA) and T is the temperature in Kelvin.
Now, we can express the volume (V) in terms of the number of molecules (N) using the density (ρ) of the gas:
V = N/ρ
Substituting this into the above equation, we get:
P = kTρ
Using the ideal gas law equation, we can express the density (ρ) in terms of the molar mass (M) and the volume (V):
ρ = M/V
Substituting this into the above equation, we get:
P = kTM/V
Finally, we can express the Boltzmann constant (k) in terms of the gas constant (R) and Avogadro's number (NA):
k = R/NA
Substituting this into the above equation, we get:
P = (R/NA)TM/V
Simplifying this expression, we get:
P = (M/3)V2C2
where C is the root-mean-square speed of the gas molecules, given by:
C = √(3RT/M)
Substituting this into the above equation, we get the final expression for pressure exerted by an ideal gas molecule:
P = (M/3V)C2
Therefore, the correct answer is option 'D'.
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Pressure exerted by an ideal gas molecule is given by the expressiona)...
Vivax Solutions
Physics
The Kinetic Theory of Gases
The Kinetic Theory of Gases has put forward a series of assumptions in order to explain what has been observed experimentally in gases.
Although, their number may vary, the core message is the same. They are as follows:
The molecules of a particular gas are identical and in random motion.
The collisions between the molecules and the between them and the walls of the container are perfectly elastic.
The volume of a molecule is negligible, compared with the volume of the container.
There are no intermolecular attractions between the molecules.
The time taken for a collision between two molecules is negligible compared with the time taken for the same between a molecule and the wall.
Based on these assumptions, a formula can be derived that connects the pressure, volume, the number of molecules, individual mass and of course, the mean velocity.
If an individual molecule collides with a wall, its momentum gets doubled.A gas molecule can move in any direction at a given time - in the x-direction, y-direction or z-direction.
Let's consider the motion in the x-direction, as shown in the animation; let the velocity be Ux
If the length of the cube, mass of the molecule and velocity are l, m and v respectively,
Momentum in the x-direction = mUx
Momentum in the -x-direction = -mUx
Change in momentum = 2mUx
Total time taken - from one end to the other and vice versa - = 2l / Ux
Rate of change in momentum = 2mu/(2l / Ux)
= mUx2/l
According to Newton's Second Law, the rate of change of momentum is the force exerted by the molecule on the wall.
Therefore, Force = [mUx / l]
Since pressure, P = force / area
Pressure on the wall, P = [mUx2/l] / l2
= mUx2 / l3
= mUx2 / V, where V is the volume of the container, THE cube.
If there are N molecules in the container,
P = m[U1x2 + U2x2 + U3x2 + ... + Unx2 ] / V
If the velocities are equal,
P = m[NUx2 ] / V P = NmUx2 / V
velocity components
Since velocity in each direction is equal,
U2 = Ux2 + Ux2 + Ux2 = 3Ux2
Ux2 = U2 / 3
U2 is called the mean square velocity. Therefore, it is written as c2̄
Ux2 = c2̄ / 3
So, P = Nmc2̄ / 3V
PV = 1/3 mNc2̄
Physics
The Kinetic Theory of Gases
The Kinetic Theory of Gases has put forward a series of assumptions in order to explain what has been observed experimentally in gases.
Although, their number may vary, the core message is the same. They are as follows:
The molecules of a particular gas are identical and in random motion.
The collisions between the molecules and the between them and the walls of the container are perfectly elastic.
The volume of a molecule is negligible, compared with the volume of the container.
There are no intermolecular attractions between the molecules.
The time taken for a collision between two molecules is negligible compared with the time taken for the same between a molecule and the wall.
Based on these assumptions, a formula can be derived that connects the pressure, volume, the number of molecules, individual mass and of course, the mean velocity.
If an individual molecule collides with a wall, its momentum gets doubled.A gas molecule can move in any direction at a given time - in the x-direction, y-direction or z-direction.
Let's consider the motion in the x-direction, as shown in the animation; let the velocity be Ux
If the length of the cube, mass of the molecule and velocity are l, m and v respectively,
Momentum in the x-direction = mUx
Momentum in the -x-direction = -mUx
Change in momentum = 2mUx
Total time taken - from one end to the other and vice versa - = 2l / Ux
Rate of change in momentum = 2mu/(2l / Ux)
= mUx2/l
According to Newton's Second Law, the rate of change of momentum is the force exerted by the molecule on the wall.
Therefore, Force = [mUx / l]
Since pressure, P = force / area
Pressure on the wall, P = [mUx2/l] / l2
= mUx2 / l3
= mUx2 / V, where V is the volume of the container, THE cube.
If there are N molecules in the container,
P = m[U1x2 + U2x2 + U3x2 + ... + Unx2 ] / V
If the velocities are equal,
P = m[NUx2 ] / V P = NmUx2 / V
velocity components
Since velocity in each direction is equal,
U2 = Ux2 + Ux2 + Ux2 = 3Ux2
Ux2 = U2 / 3
U2 is called the mean square velocity. Therefore, it is written as c2̄
Ux2 = c2̄ / 3
So, P = Nmc2̄ / 3V
PV = 1/3 mNc2̄
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Pressure exerted by an ideal gas molecule is given by the expressiona)P=MC2/Vb)P=M2/CV2c)P=M/VCd)P=MC2/3VCorrect answer is option 'D'. Can you explain this answer? for Class 11 2025 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Pressure exerted by an ideal gas molecule is given by the expressiona)P=MC2/Vb)P=M2/CV2c)P=M/VCd)P=MC2/3VCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 11 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Pressure exerted by an ideal gas molecule is given by the expressiona)P=MC2/Vb)P=M2/CV2c)P=M/VCd)P=MC2/3VCorrect answer is option 'D'. Can you explain this answer?.
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