Adiabatic Process (Equations Proof)

Mar 2020
2
0
Europe
pV^(γ)=constant (1)
p*V=n*R*T <=> (p*V/T)=n*R <=> (p*V/T)=constant
<=> p=(constant*T)/V (2)
<=> V=(constant*T)/p (3)

(1)=>(2): T*V^(γ-1)=constant (4)

(1)=>(3): p^(1-γ)*T^(γ)=constant (5)

I can't understand how we went from equation (5) to this equation: p^[(1-γ)/γ]*T=constant

It's a^(1-x)*b^(γ)=a^[(1-x)/x)]*b?

Sorry for this stupid question but I am very new to this.

Thank you in advance!
 
Apr 2015
1,238
359
Somerset, England
Remember gamma is a constant so therefore is the gamma-th root.

Given your equation 5

\(\displaystyle {P^{\left( {1 - \gamma } \right)}}{T^\gamma } = Const\)

Take the gamma-th root of each side

\(\displaystyle \sqrt[\gamma ]{{{P^{\left( {1 - \gamma } \right)}}{T^\gamma }}} = \sqrt[\gamma ]{{Const}}\)


\(\displaystyle {P^{\frac{{\left( {1 - \gamma } \right)}}{\gamma }}}T = AnotherConst\)
 
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