Question 908: Thermodynamics & KTG

Question

The initial pressure and volume of a gas is '$$\mathrm{P}$$' and '$$\mathrm{V}$$' respectively. First by isothermal process gas is expanded to volume '$$9 \mathrm{~V}$$' and then by adiabatic process its volume is compressed to '$$\mathrm{V}$$' then its final pressure is (Ratio of specific heat at constant pressure to constant volume $$=\frac{3}{2}$$)

Accepted Answer

To determine the final pressure of the gas after undergoing an isothermal expansion followed by an adiabatic compression, let's break down the problem step-by-step.
Step 1: Isothermal Expansion
During an isothermal process, the temperature remains constant. For an ideal gas, the relationship between pressure and volume is given by:
$$P_1 V_1 = P_2 V_2$$
Here, the initial pressure is $$\mathrm{P}$$ and the initial volume is $$\mathrm{V}$$. After the isothermal expansion, the volume becomes $$9 \mathrm{~V}$$ and the pressure becomes $$\mathrm{P_2}$$.
So we have:
$$\mathrm{P} \cdot \mathrm{V} = \mathrm{P_2} \cdot 9 \mathrm{V}$$
Solving for $$\mathrm{P_2}$$:
$$\mathrm{P_2} = \frac{\mathrm{P}}{9}$$
Step 2: Adiabatic Compression
During an adiabatic process, no heat is exchanged with the surroundings. The relationship between pressure and volume for an adiabatic process is given by:
$$P_2 V_2^\gamma = P_3 V_3^\gamma$$
Here, $$\gamma$$ is the adiabatic index, which is the ratio of specific heat at constant pressure to constant volume ($$=\frac{3}{2} = 1.5$$). The final volume $$V_3$$ is equal to the initial volume $$\mathrm{V}$$.
Given:
$$\mathrm{P_2} = \frac{\mathrm{P}}{9}, \quad \mathrm{V_2} = 9 \mathrm{V}, \quad \mathrm{V_3} = \mathrm{V}$$
Thus, we have:
$$\left(\frac{\mathrm{P}}{9}\right) \cdot (9 \mathrm{V})^\gamma = \mathrm{P_3} \cdot \mathrm{V}^\gamma$$
Substituting $$\gamma = 1.5$$:
$$\left(\frac{\mathrm{P}}{9}\right) \cdot (9 \mathrm{V})^{1.5} = \mathrm{P_3} \cdot \mathrm{V}^{1.5}$$
Since $$\gamma = 1.5$$, we have:
$$ (9 \mathrm{V})^{1.5} = 9^{1.5} \cdot \mathrm{V}^{1.5} = 27 \cdot \mathrm{V}^{1.5}$$
Thus:
$$ \left(\frac{\mathrm{P}}{9}\right) \cdot 27 \cdot \mathrm{V}^{1.5} = \mathrm{P_3} \cdot \mathrm{V}^{1.5} $$
Dividing both sides by $$\mathrm{V}^{1.5}$$:
$$ \left(\frac{\mathrm{P}}{9}\right) \cdot 27 = \mathrm{P_3} $$
Simplifying, we get:
$$\mathrm{P_3} = 3 \mathrm{P}$$
So, the final pressure of the gas after the adiabatic compression is
3P
.
Therefore, the correct answer is:
Option C: 3 P

Answer

6 P

Answer

27 P

Answer

3 P

Answer

9 P

Thermodynamics & KTG

Choose the correct answer: