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I have

$$VT^\frac{f}{2}=\text{constant}\tag1$$

and

$$V^{\gamma}P=\text{constant}.\tag2$$

Where $V=\text{volume}, \ P=\text{pressure} , \ T=\text{temperature},$ $f=\text{number of degrees of freedom per molecule}$ and $\gamma=\frac{f+2}{f}.$

I want to derive $(2)$ from $(1)$

Using the ideal gaslaw I have $T=\frac{PV}{Nk}$, where $N$ and $k$ are constant. Plugging this into $(1)$ I get

$$V\left(\frac{PV}{Nk}\right)^{\frac{f}{2}}=\frac{V^\frac{f+2}{2}P}{\text{constant}}=\text{constant},$$

and a constant times a constant is constant so finally

$$VT^{\frac{f}{2}}=V^{\frac{f+2}{2}}P=\text{constant}.$$

However my exponent is off, the denominator in the exponent should be $f$ but I get $2$. What am I missing?

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  • $\begingroup$ Please state reason for downvote. $\endgroup$ – Parseval Sep 11 at 14:52
  • $\begingroup$ Seems to be a very 'silly' question; if not question, atleast your response to my answer is not genuine $\endgroup$ – Tojrah Sep 13 at 12:26
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You did not put $P^{f/2}$ after the first step. From there you will get the right answer (then you will take both sides of the equation to the 2/f power).

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  • $\begingroup$ Please see my response to the answer above, from Tojrah. $\endgroup$ – Parseval Sep 11 at 14:03
  • $\begingroup$ @Parseval You asked what you are missing, you want the whole rest of the derivation worked out? I edited with the next step. $\endgroup$ – user47014 Sep 11 at 14:23
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$$V\bigg(\frac {PV}{Nk} \bigg)^{f/2}=\frac{P^{f/2} V^{1+f/2}}{constant}$$

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  • $\begingroup$ $P$ is not supposed to have any exponent in the final answer. $\endgroup$ – Parseval Sep 11 at 13:59

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