The heat capacity of one mole of an ideal gas is found to be $C_{v}=3R(1+aRT)/2$ where $a$ is a constant. The book provides the solution to this problem but there is a step in which $(1+aRT)=e^{aRt}$ How is this possible? Please explain.
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$\begingroup$ I'm guessing you are given $nC_vdT=n\frac{3}{2}R(1+aRT)dT=-PdV=-\frac{nRT}{V}dV$. Correct? $\endgroup$– Chet MillerCommented Oct 29, 2019 at 17:51
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$\begingroup$ ${}$ Which book? $\endgroup$– Qmechanic ♦Commented Oct 29, 2019 at 19:34
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1$\begingroup$ The equation you are asking about is an approximation. If you expand $e^{aRT}$ in a Taylor series about aRT=0, the first two terms are 1 + aRT. $\endgroup$– Chet MillerCommented Oct 30, 2019 at 1:24
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If $aRT$ is extremely small then $$e^{aRT} \approx 1+aRT$$
answered Nov 9, 2019 at 12:45