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

Answer:

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.

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:

$\displaystyle\frac{T_f}{T_i}=\displaystyle(\frac{V_i}{V_f})^{\frac{1}{\gamma-1}}$

$V_3=V_2(\frac{T_{12}}{T_{34}})^{3/2}=7.7L$

$V_4=V_1(\frac{T_{12}}{T_{34}})^{3/2}=1.54L$

For the cycle:

$Q_{h}=W_{12}=nRT_{12}\frac{V_2}{V_1}=5352.35J$

$W_{23}=-\Delta U=nC_v(T_{12}-T_{34})$

$Q_{c}=W_{34}=nRT_{34}\frac{V_4}{V_3}=-4014.26J$

$W_{41}=-\Delta U=nC_v(T_{34}-T_{12})=-W_{23}$

$W_{cycle}=W_{12}+W_{23}+W_{34}+W_{41}=W_{12}+W_{34}=1338.09J$