Question

One mole of a monatomic gas performs a Carnot cycle between the temperatures 400 K (400 Kelvin) and 300 K (300 Kelvin). On the upper isothermal transformation, the initial volume is 1 liter and the final colume 5 liters. Find the work performed during a cyle, and the amounts of heat exchanged with the two sources

Answer 1

The work performed during the cycle W is 1338.09J. The heat absorbed from the hot source Qh is 5352.35J. The heat released to the cold source Qc is 4014.26J.

Step-by-step explanation:

If we considered the cycle done between four points, the initial point 1 to be the point of the initial volume (V₁=1L) and the next to be the final volume on that isothermal transformation (V₂=5L with T₁₂=400K).

Using the adiabatic relationship between Volume and temperature:

`\displaystyle(T_f)/(T_i)=\displaystyle((V_i)/(V_f))^{(1)/(\gamma-1)}`

`V_3=V_2((T_(12))/(T_(34)))^(3/2)=7.7L`

`V_4=V_1((T_(12))/(T_(34)))^(3/2)=1.54L`

For the cycle:

`Q_(h)=W_(12)=nRT_(12)(V_2)/(V_1)=5352.35J`

`W_(23)=-\Delta U=nC_v(T_(12)-T_(34))`

`Q_(c)=W_(34)=nRT_(34)(V_4)/(V_3)=-4014.26J`

`W_(41)=-\Delta U=nC_v(T_(34)-T_(12))=-W_(23)`

`W_(cycle)=W_(12)+W_(23)+W_(34)+W_(41)=W_(12)+W_(34)=1338.09J`