Step 1: Understanding the Concept:
Gas laws define the physical behavior of gases by relating properties such as Pressure (\( P \)), Volume (\( V \)), and Temperature (\( T \)).
- Boyle's Law: For a fixed amount of gas at constant temperature, \( P \propto 1/V \) or \( PV = k \).
- Charles's Law: For a fixed amount of gas at constant pressure, \( V \propto T \) or \( V/T = k \).
- Gay-Lussac's Law: For a fixed amount of gas at constant volume, \( P \propto T \) or \( P/T = k \).
By combining these individual laws for a specific mass of gas where quantity \( n \) is constant, we derive the Combined Gas Law.
Step 2: Key Formula or Approach:
The combined gas law states that the ratio of the product of pressure and volume to the absolute temperature is constant for a fixed mass of gas.
\[ \frac{PV}{T} = C \]
For comparing two states of the same gas:
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]
Step 3: Detailed Explanation:
Let's analyze why option (C) is correct and others are wrong:
- In state 1, the gas has pressure \( P_1 \), volume \( V_1 \), and temperature \( T_1 \).
- In state 2, the gas has pressure \( P_2 \), volume \( V_2 \), and temperature \( T_2 \).
Since the amount of gas does not change, \( \frac{P_1V_1}{T_1} \) must equal \( \frac{P_2V_2}{T_2} \).
Option (A) is incorrect because it swaps the subscripts of volume, which violates the proportionalities of Boyle's and Charles's laws.
Option (B) is a wrong rearrangement of the Ideal Gas Law \( PV = nRT \). The correct form would be \( n = PV/RT \).
Option (D) is also a wrong algebraic derivation of the Ideal Gas Law. The correct form is \( P = nRT/V \).
Step 4: Final Answer:
The equation \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \) correctly represents the combined behavior of pressure, volume, and temperature as derived from Boyle's and Charles's laws.
Thus, the correct option is (C).