Physics: Brownian Motion and Ideal Gas Law

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Questions and Answers

What characterizes Brownian motion?

Particles move randomly in straight lines at uniform speed. (correct)
Particles move randomly in circular paths.
Particles have varying speeds in their movement.
Particles are stationary and do not move.

What happens to the speed of a particle during an elastic collision?

The speed doubles after the collision.
The speed remains constant before and after the collision. (correct)
The speed becomes zero after the collision.
The speed decreases by half after the collision.

What does the variable $n$ represent in the equation $PV = nRT$?

The number of moles of gas (correct)
The pressure of the gas
The volume of the gas
The absolute temperature in Kelvin

Which of the following statements about the collisions between molecules is correct?

Collisions are elastic, maintaining kinetic energy. (D) Signup and view all the answers

Which constant is represented by $R$ in the equation $PV = nRT$?

The universal gas constant (B) Signup and view all the answers

When a particle collides elastically, which of the following is true?

The velocities of the colliding particles may change, but their total energy is conserved. (B) Signup and view all the answers

Which of the following best describes the freedom of movement of particles exhibiting Brownian motion?

Particles move randomly in all directions without restriction. (D) Signup and view all the answers

What is the significance of $N$ in the equation provided?

The number of molecules of the gas (C) Signup and view all the answers

What is Avogadro's number ($N_A$) approximately equal to?

6.022 imes 10^{23} (B) Signup and view all the answers

In the equation $PV = nRT$, which of the following variables directly influences the pressure of a gas?

All of the above (D) Signup and view all the answers

What is the relationship between pressure and temperature according to the given equations?

Pressure increases with temperature for constant volume. (D) Signup and view all the answers

What do $P_i$ and $T_i$ represent in the context of the equations?

Initial pressure and initial temperature. (C) Signup and view all the answers

In the equations provided, what does the symbol $eta$ represent?

Constant specific to the substance. (D) Signup and view all the answers

What does the equation $rac{eta P}{rac{ ext{d}P}{ ext{d}T}}$ indicate when considering the variables?

The rate of change of pressure with respect to temperature. (D) Signup and view all the answers

Which equation correctly describes the differential change in pressure ($ ext{d}P$)?

$ ext{d}P = rac{ ext{d}T}{V} - rac{ ext{d}V}{T}$ (B) Signup and view all the answers

What is the formula for calculating the number of moles (n) of a substance?

n = N / N_A (D) Signup and view all the answers

Which of the following represents the equation of state for an ideal gas?

PV = nRT (A) Signup and view all the answers

At standard temperature and pressure (STP), what is the temperature in Kelvin?

273 K (A) Signup and view all the answers

If the molecular mass of a substance is M, what is the mass of one molecule (m')?

m' = M / N_A (D) Signup and view all the answers

In the equation P = ρHg × g × hHg, what does ρHg represent?

Density of mercury (D) Signup and view all the answers

What is the product of $8$ and $0.76$?

$6.08$ (C) Signup and view all the answers

What is the conversion of $1 ext{ atm}$ into $ ext{N/m}^2$ using the given equation?

$1.013 imes 10^5 ext{ N/m}^2$ (A) Signup and view all the answers

What would be the temperature at which the volume of a given mass of an ideal gas doubles at $27°C$, if the pressure remains constant?

$54°C$ (A) Signup and view all the answers

If the equation $P = 1 ext{ atm} = 1 imes 1.013 imes 10^5 ext{ N/m}^2$ holds, what is the numerical value of pressure in $ ext{N/m}^2$?

$1.013 imes 10^5$ (A) Signup and view all the answers

What is the value of $1.013 imes 10^5 ext{ N/m}^2$ in standard units if $1 ext{ atm}$ is approximately equal to?

$101325 ext{ N/m}^2$ (A) Signup and view all the answers

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Study Notes

Brownian Motion

Particles move randomly in straight lines at a uniform speed in all directions.
Collisions between molecules are elastic: the speed before a collision equals the speed after a collision.

Ideal Gas Law

PV = nRT
- P is pressure (in Pascals (Pa))
- V is volume (in cubic meters (m³)).
- n is the number of moles of gas.
- R is the ideal gas constant (8.314 J/mol⋅K).
- T is temperature (in Kelvin (K)).

Number of Moles

n = m/M
- m is the mass of the gas (in grams (g)).
- M is the molar mass of the gas (in grams per mole (g/mol)).
n = N/NA
- N is the number of molecules.
- NA is Avogadro's number, 6.022 × 10²³ molecules/mol.

Pressure, Volume, and Temperature Relationships

V = nRT/P
T= PV/nR
P = nRT/V

Relationship Between Pressure and Temperature

For a constant volume: The pressure and temperature of a gas are directly proportional.
- If temperature increases, pressure increases.
- If temperature decreases, pressure decreases.

Standard Temperature and Pressure (STP)

T = 0°C = 273 K
P = 1 atm = 1.013 x 10⁵ N/m²
This is the pressure exerted by a column of mercury 760 mm (0.76 m) high at 0°C.
This is equivalent to 101.3 kPa.

Example: Calculating Temperature with Doubling Volume

Given: Initial temperature (Ti) = 27°C = 300 K, Volume doubles.
Applying the Ideal Gas Law: V₁/T₁ = V₂/T₂ with V₂ = 2V₁
Solving for T₂: T₂ = (V₂/V₁)T₁ = (2V₁/V₁)T₁ = 2 * 300 K = 600 K.
The final temperature is 600 K, which is 327°C.

Mass of a Single Molecule

m’ = M/NA
- M is the molar mass
- NA is Avogadro's number
- m’ is the mass of a single molecule.

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