Gas Leak (Effusion): Solving Differential Equation

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Homework Help Overview

The discussion revolves around a differential equation related to gas effusion from a container through a small pinhole. The original poster is exploring the implications of the equation dN/dt = (Constants)*N, particularly focusing on the dimensional analysis of the exponent in the resulting exponential solution.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand why the exponent in the solution appears not to involve time, raising concerns about dimensional consistency. Other participants discuss the nature of the constants involved and their units, with some clarifying that the exponent must be dimensionless.

Discussion Status

Participants are actively engaging in clarifying the dimensional analysis of the constants in the differential equation. Some guidance has been offered regarding the units of the constants, and there is an ongoing exploration of the assumptions behind the equation.

Contextual Notes

There is a mention of the specific physical context involving gas effusion, and participants are considering the definitions and units of various parameters such as area and volume. The original poster expresses confusion about the dimensionality of the exponent, suggesting a need for further clarification.

moonman
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I'm working on a question where gas is leaking out of a container from a small pinhole. I have a differential equation dN/dt = (Constants)*N. I solved this to get an exponential. The exponent is that group of constants, which are A/2V*(kT/m)^1/2. Shouldn't the exponent have something to do with time? Because if they don't, the dimensions don't work out. Can anyone clear this up for me?
 
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dN/dt = C * N
yields
N = N0 * exp (C * t)
,where t is time

BTW, this is not "Advanced Physics" ;-)
 
You're right, that's not advanced physics, its simple DE. But that's not what I asked.
For that equation to make sense, the exponent should be dimensionless. C*t is not. That's the part I don't understand
 
N is a count ... dN/dt has units counts/sec ... C must have units 1/sec.
 
If A is area and V is volume (and yes, they are) then dimension of your constant is
m2/m3 * m/s = 1/s
and no problem.
I advise to use more precise formula
dN/dt = -1/4 n*<v>*A = -1/4 AN/V * sqrt[kT/(2*3.1416*m)]
 

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