To tackle this question, we need to analyze the two processes described for the expansion of an ideal gas. The processes are defined by the equations: 1) PV = K and 2) P = KV², where K is a positive constant. Each process will affect the final temperature, work done, and heat transfer differently. Let's break down each option systematically.
Understanding the Processes
In the first process, described by the equation PV = K, we can rearrange it to express pressure in terms of volume:
P = K/V
This indicates that as the volume increases, the pressure decreases inversely. This is a typical isothermal process for an ideal gas, where the temperature remains constant if the gas expands slowly enough.
In the second process, P = KV² suggests that pressure increases with the square of the volume. This relationship indicates a different kind of expansion, where pressure does not decrease as volume increases. Instead, it implies that the gas is being compressed or expanded in a manner that does not maintain constant temperature.
Analyzing Each Statement
- a) Final temperature in 1st will be greater than in 2nd: This statement is likely true. In the first process, if we assume an isothermal expansion, the temperature remains constant. In the second process, since pressure increases with volume, it suggests that the gas may not be expanding isothermally, leading to a higher final temperature.
- b) Final temperature in 2nd will be greater than in 1st: This is likely false based on our analysis. The first process maintains a constant temperature, while the second process may lead to a higher temperature due to the nature of the pressure-volume relationship.
- c) Work done by the gas in both processes would be equal: This is not necessarily true. The work done by the gas depends on the path taken during expansion. Since the equations governing the two processes are different, the work done will also differ.
- d) Total heat given to gas in 1st is greater than in 2nd: This statement is also likely false. In the first process, if the temperature is constant, the heat added to the system is equal to the work done. In the second process, the heat transfer could be less due to the nature of the expansion.
Final Thoughts
Based on the analysis, the most accurate statement appears to be option a: the final temperature in the first process will be greater than in the second. The reasoning hinges on the nature of the processes and how they affect the temperature and work done by the gas. Understanding these relationships is crucial in thermodynamics, especially when dealing with ideal gases and their behaviors under different conditions.
