Thermodynamics: p/V = constant

[KingAntikrist]

First of all, i want to begin with saying that this is my translation of the problem in english (the original is in romanian) so i don't know if some words are correctly used...

Homework Statement

A monoatomic ideal gas expands under the law $p=a*V$ with a = constant. Find the molar heat in this transformation.

A) 6 R
B) 2 R
C) 0.5 R
D) R
E) 3 R
F) 5 R

Homework Equations

http://img710.imageshack.us/img710/8069/asd3432re.jpg

I forgot this: $C_V=\frac{3}{2}R$

The Attempt at a Solution

http://img534.imageshack.us/img534/6503/32ef5456436tgr.jpg

Where's my mistake :-s ?

 

[hikaru1221]

From line 5 to line 6: 1/V is not p/a. Because p=aV, it should be a/p instead.

 

[KingAntikrist]

Oh yea, thanks for pointing this out...

Now I'm stuck at $\frac{T}{pV}=\frac{1}{a}$ = constant

 

[hikaru1221]

You end up at the first place, right?
Why don't you try to do something about the 2nd relevant equation you gave above? They ask for C, right?

 

[KingAntikrist]

You end up at the first place, right?
Why don't you try to do something about the 2nd relevant equation you gave above? They ask for C, right?

ye but i don't know which Q should i "use" for a p/V transformation... if it was isobar it would be $c=\frac{\nu C_p \Delta T}{\nu \Delta T}=C_p=\frac{5}{2} R$ if it was isochoric it would be 3/2 R

 

[hikaru1221]

The textbook derives C for you in special cases. This case doesn't fall into any of those, so you must derive it from the start, i.e. from the definition of the molar heat capacity, the ideal gas equation, the 1st law of thermodynamics and most importantly the law of the process

You have: $$dQ = nCdT$$.

Moreover: $$dU = dQ - dA = dQ - pdV$$.

You also have: $$dU = nC_vdT$$

Therefore: $$C = \frac{dQ}{ndT} = C_v + \frac{1}{n}\frac{pdV}{dT}$$

Now try to find pdV/dT from the 2 equations: pV/T = const and p/V = const

 

[KingAntikrist]

The textbook derives C for you in special cases. This case doesn't fall into any of those, so you must derive it from the start, i.e. from the definition of the molar heat capacity, the ideal gas equation, the 1st law of thermodynamics and most importantly the law of the process

You have: $$dQ = nCdT$$.

Moreover: $$dU = dQ - dA = dQ - pdV$$.

You also have: $$dU = nC_vdT$$

Therefore: $$C = \frac{dQ}{ndT} = C_v + \frac{1}{n}\frac{pdV}{dT}$$

Now try to find pdV/dT from the 2 equations: pV/T = const and p/V = const

I don't want to be rude, but can you solve it without using those "d's" (differentiation) because i wasn't taught at school thermodynamics with derivatives (or integrals) and i don't understand what you did over there :-s

 

[hikaru1221]

Oh I see. Then just change $$d$$ into $$\Delta$$ and you should get the same result:
$$C = \frac{\Delta Q}{n\Delta T} = C_v + \frac{1}{n}\frac{p\Delta V}{\Delta T}$$
There is nothing about differentiation here; just the notation

 

[KingAntikrist]

Oh I see. Then just change $$d$$ into $$\Delta$$ and you should get the same result:
$$C = \frac{\Delta Q}{n\Delta T} = C_v + \frac{1}{n}\frac{p\Delta V}{\Delta T}$$
There is nothing about differentiation here; just the notation

i found out that $\frac{1}{\nu}\frac{p\Delta V}{\Delta T}$ is R ... but the answer's not good :-s ...

well isn't $\frac{pV}{T}$ constant? equal to $\nu R$

 

[hikaru1221]

Yes. But not $$\frac{p\Delta V}{\Delta T}$$.

We can change it a bit: $$\frac{p\Delta V}{\Delta T}=\frac{aV\Delta V}{\Delta T}$$

We need to find the relation between $$\Delta V$$ and $$\Delta T$$

$$pV/T = nR$$ so $$V^2 = (nR/a)T$$. Therefore: $$\Delta (V^2) = (nR/a)\Delta T$$, agree?

As the volume changes slightly from $$V$$ to $$V+\Delta V$$, we have:
$$\Delta (V^2)=(V+\Delta V)^2-V^2=2V\Delta V + (\Delta V)^2$$

Because $$\Delta V << V$$, we can omit the 2nd term, and thus: $$\Delta (V^2)=2V\Delta V$$

The rest should be easy

 

[vela]

You can't assume $\Delta V \ll V$.

Draw a P-V diagram showing the path the system follows going from V₁ to V₂. The work done is given by the area under that curve, which you can calculate geometrically from the diagram.

 

[hikaru1221]

You can't assume $\Delta V \ll V$.

Draw a P-V diagram showing the path the system follows going from V₁ to V₂. The work done is given by the area under that curve, which you can calculate geometrically from the diagram.

Why not?