Answer to Question #98807 in Electrical Engineering for Hardi Amiru Sulemana

Question #98807
Show that during the expansion of a quantity of gas according to the law PV^n = constant,the heat energy transfer Q and W are connected by the expression
Q={r - n/r - 1}W .
1
Expert's answer
2019-11-18T08:25:15-0500

The work during that process can be expressed as


W=pbVbpaVan1.W=\frac{p_bV_b-p_aV_a}{n-1}.

According to the first law of thermodynamics, for a gas with degrees of freedom of molecules ii we can write:

Q=ΔU+W=i2mμRΔT+pbVbpaVan1.Q=\Delta U+W=\frac{i}{2}\frac{m}{\mu}R\Delta T+\frac{p_bV_b-p_aV_a}{n-1}.

Since

pV=mμRT,pV=\frac{m}{\mu}RT,

we can now write that


Q=ΔU+W=i2mμRΔT+mμRΔTn1= =mμRΔT(i2+1n1)=mμRΔTini+22(n1).Q=\Delta U+W=\frac{i}{2}\frac{m}{\mu}R\Delta T+\frac{\frac{m}{\mu}R\Delta T}{n-1}=\\ \space\\ =\frac{m}{\mu}R\Delta T\bigg(\frac{i}{2}+\frac{1}{n-1}\bigg)=\frac{m}{\mu}R\Delta T\cdot\frac{in-i+2}{2(n-1)}.


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