How is that homogenous with respect to units?

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Discussion Overview

The discussion revolves around the concept of homogeneity with respect to units in the context of the universal gas law equation, PV = nRT. Participants explore the meaning of homogeneity, particularly in relation to the dimensional consistency of the equation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the term "homogeneous" and its application to the gas law equation.
  • One participant suggests that "homogeneous" refers to the uniformity of variables (P, V, T) throughout the medium before reaching thermodynamic equilibrium.
  • Another participant clarifies that the question pertains to the homogeneity of units, asking how to demonstrate that the units match on both sides of the equation.
  • A participant proposes writing the dimensions in terms of [M], [L], and [T] for each quantity to verify dimensional consistency.
  • One participant asserts that the equation PV = nRT is dimensionally correct, providing a breakdown of the units involved for pressure, volume, and temperature.

Areas of Agreement / Disagreement

There is no consensus on the interpretation of "homogeneous" in this context, with some participants focusing on the uniformity of variables and others on the dimensional analysis of the equation. The discussion remains unresolved regarding the initial question posed by the original poster.

Contextual Notes

Participants have not fully clarified the assumptions behind their interpretations of homogeneity, nor have they reached a definitive agreement on the meaning of the question regarding units.

amimeera
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how is that homogenous with respect to units?
i can't get it!
 
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I don't understand the question. "Homogenous" means "the same everywhere." This equation represent the "universal gas law" which by its title implies it is "the same everywhere." Somehow, I don't think that this is what the question is after. Can you give us the full question?
 
amimeera said:
how is that homogenous?
i can't get it!

The scalar field describing the gas property (in this case temperature) is homogenous.

Rev Prez
 
My guess is that the word "homogeneous" here means that the variables in the equation (P,V,T) are the same thoughout the medium, and do not vary from point to point as would happen before they reach thermodynamic equilibrium.
 
sorry
homogenous with respect to units!
 
It looks weird.It makes no sense with "homogeneity",even in Euler sense.

Daniel.
 
amimeera said:
sorry
homogenous with respect to units!
So... is the issue how to show that the units match on both sides?

If so: What are the standard units of each quantity?
 
Doc Al said:
So... is the issue how to show that the units match on both sides?
I'm pretty sure it is...

To the OP : Write the dimensions in terms of [M], [L] and [T] for each quantity on both sides and check that the final dimensions are the same.

[P] (pressure) = [force] / [area] = [mass] [acceleration] [L^-2] = ([M] [length] / [time^2]) * [L^-2] = [M] [L^-1] [T^-2]

Do the others similarly (and get the units for R correct)
 
amimeera:

What I'm guessing you meant to ask is: Is the equation PV = nRT dimensionally correct? In other words, do the "units" on both sides of the equation match?

The answer is: Yes.

In SI units we have:

P is in Pascals. 1 Pa = 1 N m^-2 = 1 kg m^-1 s^-2
V is in cubic metres (m^3).

Therefore PV has units kg m^2 s^-2, which is the same as Joules. Another way to say that is that the dimension of PV is the same as energy.

n has no units.
R is the gas constant, with units J K^-1.
Temperature is in Kelvin (K).

nRT therefore has units of Joules, or dimensions of energy.

Since PV and nRT both have dimensions of energy, the equation PV=nRT is dimensionally correct.
 

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