What pressure must be applied to a given sample of a gas in order to compress it to three- fourth of its original volume ?

Explanation

To solve the problem of determining the pressure required to compress a gas to three-fourths of its original volume, we can use Boyle's Law. Here's a step-by-step solution:

### Step 1: Understand Boyle's Law
Boyle's Law states that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, it can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
where:
- \( P_1 \) = initial pressure
- \( V_1 \) = initial volume
- \( P_2 \) = final pressure
- \( V_2 \) = final volume

### Step 2: Define the Variables
Let:
- \( P_1 = P \) (the initial pressure)
- \( V_1 = V \) (the initial volume)
- \( V_2 = \frac{3}{4} V \) (the final volume after compression)

### Step 3: Substitute the Known Values into Boyle's Law
Using Boyle's Law, we can substitute the known values into the equation:
\[ P_1 V_1 = P_2 V_2 \]
Substituting the values we defined:
\[ P V = P_2 \left(\frac{3}{4} V\right) \]

### Step 4: Solve for \( P_2 \)
To find \( P_2 \), we rearrange the equation:
\[ P_2 = \frac{P V}{\frac{3}{4} V} \]

### Step 5: Simplify the Equation
The volume \( V \) cancels out:
\[ P_2 = \frac{P}{\frac{3}{4}} \]
This can be rewritten as:
\[ P_2 = P \times \frac{4}{3} \]

### Step 6: Final Result
Thus, the pressure that must be applied to compress the gas to three-fourths of its original volume is:
\[ P_2 = \frac{4}{3} P \]