Brownian Motion Formula, Definition, Solved Examples

Brownian Motion Formula: Brownian Movement refers to the unpredictable motion of particles within a fluid. This phenomenon is characterized by the erratic, zigzag trajectory of a particle, typically discernible through a high-power ultra-microscope. It can be image as the unregulated and irregular displacement of particles in a fluid, arising from continuous interactions with swiftly moving molecules.

The degree of random motion in a particle is generally more pronounced in smaller particles, less viscous liquids, and elevated temperatures. As these particles traverse, they engage in collisions with each other. Brownian motion provides insight into the haphazard movement of minuscule particles suspended in fluids.

Brownian Motion Formula

The calculation of Brownian motion involves the use of a parameter called the diffusion constant, denoted by the symbol D. The formula for D is derived as the ratio of the product of the gas constant and temperature to the product of six times pi, Avogadro’s number, the viscosity of the fluid, and the radius of the particle. Notably, D is dimensionless, as it represents the ratio of identical quantities, devoid of a dimensional formula.

Here are the rewritten formulas:

D= RT ​/ 6πrηN _a

or

D= k _B ​ T ​/ 6πrη

Where,  D is the diffusion constant,

• k _B ​ is the Boltzmann constant ( 1.381×10 ⁻²³ J/K),
• R is the gas constant ( 8.314J K ⁻¹ mol ⁻¹ ),
• T is the temperature of the surroundings,
• π is a constant ( 3.14),  r is the radius of the Brownian particle,
• η is the fluid viscosity,
• N _a ​ is Avogadro’s number ( 6.06×10 ²³ mol ⁻¹ ).

Brownian Motion Formula Solved Examples

Example 1: Calculate the diffusion constant (D) for a Brownian particle with a radius (r) of 2 m, fluid viscosity (η) of 0.056 Pa, and temperature (T) of 300 K.

D= k B ​ T / 6πrη  ​ = (1.381×10 ⁻²³ ×300) /  (6×3.14×2×0.056)

​ =1.96×10 ⁻²¹

Example 2: Find the diffusion constant (D) for a Brownian particle with a radius (r) of 3 m, fluid viscosity (η) of 0.068 Pa, and temperature (T) of 250 K.

D= k B ​ T / 6πrη ​ = (1.381×10 ⁻²³ ×250)/ (6×3.14×3×0.068) ​

=8.98×10 ⁻²²

Example 3: Determine the temperature (T) of the surroundings if the diffusion constant (D) for a Brownian particle is 7.5×10 ⁻²² , the radius (r) is 2.5 m, and fluid viscosity (η) is 0.087 Pa s.

T= ​ 6πDrη / k B ​

= (6×3.14×7.5×10 ⁻²² ×2.5×0.087) ​/ (1.381×10 ⁻²³ )

=222.5K

Example 4: Calculate the diffusion constant (D) for a Brownian particle with a radius (r) of 1.5 m, fluid viscosity (η) of 0.042 Pa, and temperature (T) of 325 K.

D= k B ​ T / 6πrη  ​

= (1.381×10 ⁻²³ ×325) /(6×3.14×1.5×0.042)

=1.85×10 ⁻²¹

Example 5: Find the diffusion constant (D) for a Brownian particle with a radius (r) of 2.8 m, fluid viscosity (η) of 0.075 Pa, and temperature (T) of 275 K.

D= k B ​ T / 6πrη  ​

= (1.381×10 ⁻²³ ×275) /  (6×3.14×2.8×0.075)​

=7.21×10 −22

Example 6: Determine the temperature (T) of the surroundings if the diffusion constant (D) for a Brownian particle is 4.2×10 ⁻²² , the radius (r) is 2.0 m, and fluid viscosity (η) is 0.055 Pa s.

K T= ​ 6πDrη / k B​

= (6×3.14×4.2×10 ⁻²² ×2.0×0.055) / (1.381×10 ⁻²³ ) ​

=232.8K

The Brownian Motion Formula, D= k _B ​ T/ 6πrη  ​ , encapsulates the essence of particle diffusion in a fluid, where the diffusion constant (  D) is determined by factors such as temperature ( T), particle radius (  r), and fluid viscosity (  η). This dimensionless formula, incorporating the Boltzmann constant ( k _B ​ ), provides a concise representation of the stochastic motion observed in Brownian particles.

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