Question

An ideal gas has 𝑛 moles and is monatomic with 𝛾 = 5/3. It is contained in a piston cylinder and undergoes a Carnot engine cycle between isotherms at 𝑇𝐿 and 𝑇𝐻, between states labelled 𝑎 to 𝑑 as shown on the PV diagram. The values shown on the axes are an example. During the process bcd (from b to c and then c to d) the gas does work 𝑊𝑏𝑐𝑑. The internal energy in state b is 𝑈.Derive an expression for the ratio 𝑊𝑏𝑐𝑑/𝑈𝑏 in terms of 𝑇𝐿 , 𝑇𝐻, 𝑉𝑎, and 𝑉.

Answer 1

$\gamma = \dfrac53$

number of moles = n

sample carnot cycle

At constant T,

$W_{bcd }= nRT\ln{\dfrac{V_a}{V}}$

$U_b$ = 0

$\dfrac WU = 0$

However between T_L and T_H,

$\dfrac{W_{bcd}}{U_b}= \dfrac{T_H-T_L}{T_H} \ln\dfrac{V_a}{V}$