Question

Show four steps of Carnot engine in P – V graph write the equation of each step and obtain

the work done by the system. Also obtain the efficiency of a Carnot engine.​

Answer 1

Answer:

Step 1:

Isothermal expansion: The gas is taken from P1, V1, T1 to P2, V2, T2. Heat Q1 is absorbed from the reservoir at temperature T1. Since the expansion is isothermal, the total change in internal energy is zero, and the heat absorbed by the gas is equal to the work done by the gas on the environment, which is given as:

W1→2=Q1=μ×R×T1×lnv2v1

Step 2:

Adiabatic expansion: The gas expands adiabatically from P2, V2, T1 to P3, V3, T2.

Here, work done by the gas is given by:

W2→3=μRγ−1(T1−T2)

Step 3:

Isothermal compression: The gas is compressed isothermally from the state (P3, V3, T2) to (P4, V4, T2).

Here, the work done on the gas by the environment is given by:

W3→4=μRT2lnv3v4

Step 4:

Adiabatic compression: The gas is compressed adiabatically from the state (P4, V4, T2) to (P1, V1, T1).

Here, the work done on the gas by the environment is given by:

W4→1=μRγ−1(T1−T2)

Carnot Engine

Hence, the total work done by the gas on the environment in one complete cycle is given by:

W=W1→2+W2→3+W3→4+W4→1W=μRT1lnv2v1−μRT2lnv3v4

Netefficiency=NetworkdonebythegasHeatabsorbedbythegas

Netefficiency=WQ1=Q1−Q2Q1=1−Q2Q1=1−T2T1lnv3v4lnv2v1

Since the step 2–>3 is an adiabatic process, we can write T1V2Ƴ-1 = T2V3Ƴ-1

Or,

v2v3=(T2T1)1γ−1

Similarly, for the process 4–>1, we can write

v1v2=(T2T1)1γ−1

This implies,

v2v3=v1v2

So, the expression for net efficiency of carnot engine reduces to:

Netefficiency=1−T2T1