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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
2019-11-18T08:25:15-0500

The work during that process can be expressed as

"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 "i" we can write:

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

Since

"pV=\\frac{m}{\\mu}RT,"

we can now write that

"Q=\\Delta U+W=\\frac{i}{2}\\frac{m}{\\mu}R\\Delta T+\\frac{\\frac{m}{\\mu}R\\Delta T}{n-1}=\\\\\n\\space\\\\\n=\\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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