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Equal molecules of two gases are in ther...

Equal molecules of two gases are in thermal equilibrium. If `P_(a), P_(b) and V_(a),V_(b)` are their respective pressures and volumes, then which of the following relation is true?

A

`P_(a) != P_(b) , V_(a) = V_(b)`

B

`P_(a) = P_(b) , V_(a) != V_(b)`

C

`(P_a)/(V_a) = (P_b)/(V_b)`

D

`P_(a)V_(a) = P_(b)V_(b)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the relationship between the pressures and volumes of two gases (A and B) that are in thermal equilibrium. ### Step-by-Step Solution: 1. **Understanding Thermal Equilibrium**: - When two gases are in thermal equilibrium, they are at the same temperature. Therefore, we can denote the temperatures of gas A and gas B as \( T_A = T_B \). 2. **Applying Boyle's Law**: - Boyle's Law states that for a given mass of gas at constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). Mathematically, this can be expressed as: \[ P \propto \frac{1}{V} \quad \text{(at constant temperature)} \] - This implies that: \[ PV = \text{constant} \] 3. **Setting Up the Relation**: - Since both gases A and B are at the same temperature, we can apply Boyle's Law to both gases: \[ P_A V_A = P_B V_B \] - This equation indicates that the product of pressure and volume for gas A is equal to the product of pressure and volume for gas B. 4. **Conclusion**: - The correct relation that holds true for the two gases in thermal equilibrium is: \[ P_A V_A = P_B V_B \] ### Final Answer: The relation that is true for the two gases in thermal equilibrium is: \[ P_A V_A = P_B V_B \]
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