An ideal gas is taken through a complete cycle in three steps: adiabatic expansion with work equal to$\mathbf{125}\mathbf{}\mathbf{J}$, isothermal contraction at$\mathbf{325}\mathbf{}\mathbf{K}$, and increase in pressure at constant volume.

(a) Draw a p-V diagram for the three steps

. (b) How much energy is transferred as heat in step3, and

(c) is it transferred to or from the gas?

In a process,if a system after going through multiple states comes back to its initial state, then the process is called a cyclic process. For an ideal gas, the internal energy remains conserved in a cyclic process.

1. The work done in adiabatic expansion is$\text{W}=125\text{\hspace{0.17em}J}$
2. The constant temperature of isothermal contraction is$\text{T}=325\text{K}$

For the three steps of the cyclic process- the adiabatic expansion, the isothermal contraction and the increase in pressure at constant volume, the P-V diagram is shown below-

Step 3 is of increasing pressure at a constant volume. So, all the heat exchange that takes place between system and surrounding will be used in changing the internal energy of the system. In the step 1 of the process (Adiabatic expansion), the change in internal energy of the system is$-125\text{\hspace{0.17em}J}$ . In step 2 (isothermal contraction), the change in internal energy is zero. As the process is cyclic, the internal energy change in step 3 must be .$125\text{\hspace{0.17em}J}$

So, the equation of the first law of thermodynamics will become-

$\begin{array}{l}Q=\Delta E_{\mathrm{int}}+W\\ 0=\Delta E_{\mathrm{int}}+W\\ W=-\Delta E_{\mathrm{int}}\end{array}$

As, work done is negative, the change in internal energy must be positive.

$\Delta E_{\mathrm{int}}=125\text{\hspace{0.17em}J}$

c.

As the pressure is increased at constant volume, the internal energy of the system must be increased, to maintain the constant volume of the gas. This means that the heat is being transferred to the gas.

The work done in step 3 of the process is $125\text{\hspace{0.17em}J}$and this is done by giving heat energy to the gas.