# Help with rearranging equations for gas laws

• talebone
In summary, the equation of state used to model a fixed amount of a real gas is given by (p + a/v2)(V - b) = ZT, where Z, a, and b are constants and P, V, and T represent pressure, volume, and temperature respectively. The dimensions of b are L3 and the dimensions of a are ML5/T2. The constant Z can be expressed in terms of the unit J/K. In order to solve these types of problems, it is helpful to use dimensional analysis and temporarily change quantity labels to avoid confusion.
talebone

## Homework Statement

An equation of state that has been used to model the behaviour of a fixed amount of a real gas is

(p + a/v2) (V - b) = ZT

In the equation, Z, a and b are constants, and P represents pressure, V represents volume, and T represents temperature. The constant Z depends on the amount of gas.
Consider what the dimensions of a and b are in terms of the fundamental quantities of mass (M), length (L) and time (T).

Q1
The dimensions of b are
A L3
B L6
C M L -1T-2.
D M-1 L T2 .Q2
The dimensions of a are
A L6.
B M L5 T-2.
C M L-I T-2.
D M L-5 T-1.

Q3
The value of the constant Z could be expressed in terms of the unit
A W
B W N s-l.
C N JK-1.
D J K-1

V = cm3
d = m/v
P = N/m2
PV = nRT?
W = FD

## The Attempt at a Solution

Ok. I'll be honest these questions are beyond me. At first glance I am tempted to skip it. But then with what little knowledge I had, I gave it a shot.

Q1
For one I know V = L3. Ok, we'll start there. If I'm finding b, that means L is going to be on top. I can't show it in working, but just looking at the equation, I'm guessing it. So it's either A or B.

Q2
Again it looks like L will be on top. So that leaves A or B.

Q3
Immediately it looks like K will be under. So that leaves C or D. I look at the answer I see Joules which is work (=FD), which has Newtons in it, so I don't think the ans is C. Therefore it must be D.

As you can see, I have no idea how to do these type of questions. I did a google search and found a term 'Dimensional Analysis', and apparently P = M/LT2. However, this still does not allow me to answer the question. Please if a kind soul can help me out, or is there a website you can direct me which would explain these kind of questions?

Thanks for your time, appreciate it.

You can only sum terms with the same dimensions.
You can check your results by completing the dimensional analysis to see if the dimensions on each side match up.

Notation is case-sensitive so "v" is a different variable to "V". You have only defined one of them.
I'm guessing they are both supposed to be upper case.

example: if we have a relation ##y=a(x+b)## then,
if [x] = L then =L so that [x+b]=L ...
if [y] = [force] = MLT-2 then [a] = MT-2

Last edited:
thanks Simon, got it! (yes you are right the two V's are meant to be capital)

Q1
b has to be L3 according to the rule of sums.

Q2
pressure = F/A = N/m2 = kg m/s2 x 1/m2 = kg/ms2 = M/LT2

so M/LT2 + a/(L3)2 then a = ML5/T2

Q3
(M/LT2)(L3) = ZT
ML2/T2 = ZT

work = FD = Nm = kgm/s2 x m kgm2/s2 = ML2/T2 = J

Z = J/K

talebone said:
thanks Simon, got it! (yes you are right the two V's are meant to be capital)

Q1
b has to be L3 according to the rule of sums.

Q2
pressure = F/A = N/m2 = kg m/s2 x 1/m2 = kg/ms2 = M/LT2

so M/LT2 + a/(L3)2 then a = ML5/T2

Q3
(M/LT2)(L3) = ZT
ML2/T2 = ZT

work = FD = Nm = kgm/s2 x m kgm2/s2 = ML2/T2 = J

Z = J/K

Very nice job. The only thing I would change would be to recognize in your Q3 analysis, the rhs's of the first two equations should be ZK, not ZT. You have been using the symbol T for time, not temperature.

(M/LT2)(L3) = [ZT] to distingush between T := "dimensions of time" and [T] := "dimensions of temperature". Without the square brackets, the statement is false. :)

As Chestermillar points out, it is good practice to temporarily change a quantity label when it is the same as a dimension label... just while you are doing dimensional analysis. The unit label is handy so [K] would mean "dimensions of Kelvins".

One way to figure if the math you have written makes sense is to translate it into words.

That said - good work!

## What are the gas laws?

The gas laws are a set of mathematical equations that describe the behavior of gases under different conditions of temperature, pressure, and volume.

## Why is it important to rearrange gas law equations?

Rearranging gas law equations allows us to manipulate and solve for different variables, making it easier to understand and predict the behavior of gases.

## What are some common gas law equations?

Some common gas law equations include Boyle's Law (P1V1=P2V2), Charles's Law (V1/T1=V2/T2), and Avogadro's Law (V1/n1=V2/n2).

## How do I rearrange gas law equations?

To rearrange gas law equations, you can use algebraic principles such as multiplication, division, addition, and subtraction to isolate and solve for the desired variable.

## What units should be used in gas law equations?

The units used in gas law equations depend on the specific gas law being used. However, it is important to make sure that all units are consistent and in the proper SI units (e.g. pressure in Pascals, volume in cubic meters, temperature in Kelvins).

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