Calculating Final Pressure in Gas Compression

This problem involves finding the final pressure of a gas after compression, given initial and final conditions of volume and temperature. We can use the Combined Gas Law, which relates pressure, volume, and temperature for a fixed amount of gas.

Gas Law Application

The Combined Gas Law is expressed as:

$ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $

Where:

• $P_1$ is the initial pressure
• $V_1$ is the initial volume
• $T_1$ is the initial absolute temperature (in Kelvin)
• $P_2$ is the final pressure
• $V_2$ is the final volume
• $T_2$ is the final absolute temperature (in Kelvin)

Temperature Conversion

First, convert the given temperatures from Celsius to Kelvin:

• Initial Temperature: $T_1 = 27^\circ\text{C} + 273.15 = 300.15 \text{ K}$
• Final Temperature: $T_2 = 77^\circ\text{C} + 273.15 = 350.15 \text{ K}$

For simplicity in typical exam calculations, we can often approximate $T_1 = 300 \text{ K}$ and $T_2 = 350 \text{ K}$.

Pressure Calculation

We need to find the final pressure ($P_2$). Rearranging the Combined Gas Law formula:

$ P_2 = P_1 \times \frac{V_1}{V_2} \times \frac{T_2}{T_1} $

Substitute the given values:

• $P_1 = 2 \text{ atm}$
• $V_1 = 60 \text{ cm}^3$
• $V_2 = 20 \text{ cm}^3$
• $T_1 \approx 300 \text{ K}$
• $T_2 \approx 350 \text{ K}$

Now, calculate $P_2$:

$ P_2 = 2 \text{ atm} \times \frac{60 \text{ cm}^3}{20 \text{ cm}^3} \times \frac{350 \text{ K}}{300 \text{ K}} $

Simplify the expression:

$ P_2 = 2 \times 3 \times \frac{350}{300} \text{ atm} $

$ P_2 = 6 \times \frac{35}{30} \text{ atm} $

$ P_2 = 6 \times \frac{7}{6} \text{ atm} $

$ P_2 = 7 \text{ atm} $

The final pressure is 7 atmospheric pressure.