Which of the following does not show the relationship between Boyle's law and Charles' law?

Options

A. `P_1/P_2`
B. `PV = K`
C. `P_2/P_1 = V_1/V_2`
D. `V_2/V_1 = P_1/P_2 xx T_2/T_1`

Answer

To solve the question "Which of the following does not show the relationship between Boyle's law and Charles' law?", we will follow these steps:

### Step 1: Understand Boyle's Law
Boyle's Law states that the pressure (P) of a gas is inversely proportional to its volume (V) when the temperature is held constant. This can be mathematically expressed as:
\[ P \propto \frac{1}{V} \]
or
\[ PV = k \]
where \( k \) is a constant.

### Step 2: Understand Charles' Law
Charles' Law states that the volume (V) of a gas is directly proportional to its absolute temperature (T) when the pressure is held constant. This can be mathematically expressed as:
\[ V \propto T \]
or
\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} \]

### Step 3: Establish Relationships
From Boyle's Law, we can derive the relationship between pressure and volume:
\[ \frac{P_1}{P_2} = \frac{V_2}{V_1} \]
From Charles' Law, we can derive the relationship between volume and temperature:
\[ \frac{V_1}{V_2} = \frac{T_1}{T_2} \]

### Step 4: Combine the Laws
We can combine these two laws to find a relationship that incorporates pressure, volume, and temperature. By substituting the relationships derived from Boyle's Law into the equation from Charles' Law, we can express:
\[ \frac{V_2}{V_1} = \frac{P_1 \cdot T_1}{P_2 \cdot T_2} \]

### Step 5: Analyze the Options
Now, we need to analyze the options provided in the question to identify which one does not represent a valid relationship between Boyle's Law and Charles' Law.

### Conclusion
After evaluating the options, we find that **option 4** does not show a valid relationship between Boyle's Law and Charles' Law. Therefore, option 4 is the correct answer.

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