Calculate the pressure (in atm) required to compress a gas adiabatically at atmospheric pressure to one third of ite volume. Given `gamma=1.47`

Explanation

To solve the problem of calculating the pressure required to compress a gas adiabatically from atmospheric pressure to one third of its volume, we can follow these steps:

### Step 1: Identify Given Values
- Initial pressure, \( P_i = 1 \) atm (atmospheric pressure)
- Final volume, \( V_f = \frac{V_i}{3} \) (the gas is compressed to one third of its initial volume)
- The adiabatic index, \( \gamma = 1.47 \)

### Step 2: Use the Adiabatic Condition
For an adiabatic process, the relationship between pressure and volume is given by the equation:
\[ P_i V_i^\gamma = P_f V_f^\gamma \]
We can rearrange this equation to express \( P_f \):
\[ P_f = P_i \left( \frac{V_i}{V_f} \right)^\gamma \]

### Step 3: Substitute the Values
Since \( V_f = \frac{V_i}{3} \), we can find \( \frac{V_i}{V_f} \):
\[ \frac{V_i}{V_f} = \frac{V_i}{\frac{V_i}{3}} = 3 \]

Now substituting the values into the equation for \( P_f \):
\[ P_f = P_i \left( 3 \right)^\gamma \]
Substituting \( P_i = 1 \) atm and \( \gamma = 1.47 \):
\[ P_f = 1 \times 3^{1.47} \]

### Step 4: Calculate \( 3^{1.47} \)
Using a calculator or logarithmic tables, we find:
\[ 3^{1.47} \approx 5.196 \]

### Step 5: Calculate \( P_f \)
Thus, the final pressure is:
\[ P_f \approx 5.196 \text{ atm} \]

### Conclusion
The pressure required to compress the gas adiabatically to one third of its volume is approximately:
\[ P_f \approx 5 \text{ atm} \]