- #1

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## Homework Statement

When a gas undergoes an adiabatic compression or expansion (it neither gains nor loses heat), then the rate of change of pressure, with respect to the volume, varies directly with the pressure and inversely with the volume. If the pressure is p N/m² when the volume is v m³ and the initial pressure and initial volume are p

_{0}N/m² and v

_{0}m³, respectively, show that pv

^{k}= p

_{0}v

_{0}

^{k}.

**2. The attempt at a solution**

[tex]\frac{dp}{dv} = \frac{p}{v}[/tex]

[tex]\frac{p}{dp} = \frac{v}{dv}[/tex]

[tex]\int \frac{p}{dp} = \int \frac{v}{dv}[/tex]

[tex]\ln |p| = \ln |v| + \bar{c}[/tex]

[tex]e^{\ln v + \bar{c}} = p[/tex]

[itex]p = Cv^k[/itex] where [itex]C = e^{\bar{c}}[/itex]

[itex]p_0 = Cv_0^k[/itex]

So, it appears that p0/v0

^{k}= p/v

^{k}= C, but I'm not sure how to obtain that pv

^{k}= p

_{0}v

_{0}

^{k}.

Thank you in advance.