# Equation for reversible adiabatic expansion

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.

• I'm guessing you are given $nC_vdT=n\frac{3}{2}R(1+aRT)dT=-PdV=-\frac{nRT}{V}dV$. Correct? – Chet Miller Oct 29 '19 at 17:51
• ${}$ Which book? – Qmechanic Oct 29 '19 at 19:34
• 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. – Chet Miller Oct 30 '19 at 1:24

If $$aRT$$ is extremely small then $$e^{aRT} \approx 1+aRT$$