Equation for Boyle's law is

Options

A. `(dP)/(P) = - (dV)/(V)`
B. `(dP)/(P) = + (dV)/(V)`
C. `(d^(2)P)/(P) = - (dV)/(dt)`
D. `(d^(2)P)/(P) = + (d^(2)V)/(dt)`

Explanation

To derive the equation for Boyle's Law, we start from the ideal gas equation and apply the principles of Boyle's Law. Here’s a step-by-step solution:

### Step 1: Understand Boyle's Law
Boyle's Law states that at constant temperature, the pressure (P) of a gas is inversely proportional to its volume (V). This means that if the volume decreases, the pressure increases, and vice versa.

### Step 2: Start with the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- P = pressure of the gas
- V = volume of the gas
- n = number of moles of the gas
- R = universal gas constant
- T = absolute temperature

### Step 3: Keep Temperature Constant
For Boyle's Law, we keep the temperature (T) constant. This means that the product \( nRT \) is a constant. We can denote this constant as \( k \):
\[ PV = k \]

### Step 4: Express the Inverse Relationship
From the equation \( PV = k \), we can express pressure in terms of volume:
\[ P = \frac{k}{V} \]
This shows that pressure is inversely proportional to volume.

### Step 5: Differentiate the Equation
To further analyze the relationship, we can differentiate the equation:
1. Start with \( PV = k \)
2. Differentiate both sides:
\[ d(PV) = d(k) \]
Since \( k \) is constant, \( d(k) = 0 \).

Using the product rule for differentiation:
\[ P \, dV + V \, dP = 0 \]

### Step 6: Rearranging the Differentiated Equation
Rearranging the differentiated equation gives us:
\[ P \, dV = -V \, dP \]
Dividing both sides by \( PV \):
\[ \frac{dP}{P} = -\frac{dV}{V} \]

### Conclusion
This final equation \( \frac{dP}{P} = -\frac{dV}{V} \) is the mathematical representation of Boyle's Law, indicating that the relative change in pressure is equal to the negative relative change in volume.

### Final Equation
Thus, the equation for Boyle's Law is:
\[ P_1 V_1 = P_2 V_2 \]
or in differential form:
\[ \frac{dP}{P} = -\frac{dV}{V} \]

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