To tackle the problem from HC Verma regarding the cyclic process of an ideal gas, we need to analyze the given information and apply the principles of thermodynamics. The process involves a gas that undergoes changes in volume and temperature, and we want to find the heat absorbed by the gas during this cycle.
Understanding the Process
In the cyclic process described, we have a gas that moves through points a, b, c, and d in a volume-temperature diagram. The key temperatures at points b and c are given as 500 K and 300 K, respectively. Since the gas is ideal, we can use the ideal gas law and the first law of thermodynamics to find the heat absorbed.
Key Concepts
- First Law of Thermodynamics: This states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W): ΔU = Q - W.
- Internal Energy of an Ideal Gas: For an ideal gas, the change in internal energy depends only on the temperature change and can be expressed as ΔU = nC_vΔT, where n is the number of moles, C_v is the molar heat capacity at constant volume, and ΔT is the change in temperature.
Calculating Heat Absorbed
Since the process is cyclic, the internal energy change over one complete cycle is zero (ΔU = 0). Therefore, the heat absorbed during the process can be calculated by considering the work done and the changes in temperature at specific points.
Steps to Calculate Heat Absorbed
- Identify the temperature changes: From point b (500 K) to point c (300 K), the gas cools down.
- Calculate the change in internal energy (ΔU) for the cooling process:
- ΔT = T_c - T_b = 300 K - 500 K = -200 K.
- Using ΔU = nC_vΔT, we need the value of C_v for an ideal gas. For a diatomic gas, C_v is typically around 5R/2, and for a monatomic gas, it is 3R/2. Assuming we have a diatomic gas, C_v = 5/2 R.
- Substituting values: ΔU = 2 moles * (5/2 R) * (-200 K).
- Calculate the work done (W) during the process. This can be determined from the area under the curve in the V-T diagram or using the pressure-volume relationship, depending on the nature of the process (isothermal, isobaric, etc.).
- Finally, use the first law of thermodynamics to find the heat absorbed: Q = ΔU + W.
Final Thoughts
By following these steps, you can systematically calculate the heat absorbed by the gas during the cyclic process. Remember, the specific values for C_v and the work done will depend on the type of gas and the nature of the process between the points. If you have the specific values for the work done or the type of gas, you can plug those into the equations to find the exact amount of heat absorbed.
