Introduction to Boyle's Law and Ideal Gas Behavior Under Constant Temperature

When dealing with gases, one fundamental relationship that often comes into play is Boyle's Law. According to Boyle's Law, the pressure of a given mass of gas at a constant temperature is inversely proportional to its volume. This means that as the volume of a gas increases, its pressure decreases, and vice versa, provided the temperature remains constant.

Boyle's Law and Its Mathematical Representation

The mathematical expression of Boyle's Law is given by the equation:

P1 × V1 P2 × V2

Here:

P1 represents the initial pressure, V1 represents the initial volume, P2 represents the final pressure, V2 represents the final volume.

In the given problem, we have a gas that occupies a volume of 400 mm at a pressure of 704 mm. We need to determine the pressure if the volume doubles while maintaining a constant temperature.

Applying Boyle's Law to the Given Problem

Given the initial conditions:

P1 704 mm V1 400 mm V2 2 × V1 800 mm

By rearranging the equation for Boyle's Law, we can find P2:

t[P_{2} frac{P_{1} times V_{1}}{V_{2}}]

Substituting the known values:

t[P_{2} frac{704 text{ mm} times 400 text{ mm}}{800 text{ mm}}]

Calculating P2:

t[P_{2} frac{281600 text{ mm}^{2}}{800 text{ mm}} 352 text{ mm}]

Thus, the pressure when the volume doubles at the same temperature will be 352 mm (assuming consistent units).

Exploring the Ideal Gas Law for Complex Scenarios

While Boyle's Law is a good starting point, the Ideal Gas Law expands our understanding by considering the relationship between pressure, volume, and temperature. For a more detailed scenario, we can apply the Ideal Gas Law:

t[frac{P_1V_1}{T_1} frac{P_2V_2}{T_2}]

In this example, assume the initial conditions:

P1 704 mm Hg V1 400 mm3 T1 293 K (20°C) V2 800 mm3 P2 and T2 are unknown.

Using the Ideal Gas Law, we can find T2 and subsequently P2:

t[frac{T_2}{T_1} left(frac{V_2}{V_1}right)^{-0.4}]

Calculating T2:

t[T_2 293 text{ K} times (frac{800 text{ mm}^3}{400 text{ mm}^3})^{-0.4} 293 text{ K} times 2^{-0.4} 293 text{ K} times 0.786 231.298 text{ K}]

Using the same Ideal Gas Law for P2:

t[P_2 P_1 left(frac{V_1}{V_2}right) left(frac{T_2}{T_1}right) 704 text{ mm Hg} times frac{400 text{ mm}^3}{800 text{ mm}^3} times frac{231.298 text{ K}}{293 text{ K}}]

Calculating P2:

t[P_2 704 text{ mm Hg} times 0.5 times 0.79 278.6 text{ mm Hg}]

Therefore, in this complex scenario, the pressure would be approximately 278.6 mm Hg if the volume doubles and temperature is considered.

Conclusion

Understanding Boyle's Law and the Ideal Gas Law is essential for solving problems related to gas behavior under different conditions. The choice between using these laws depends on the specific conditions of the problem, such as whether volume is constrained or can freely expand, and whether temperature needs to be considered.

Key Takeaways

Boyle's Law: P1 × V1 P2 × V2 Ideal Gas Law: (frac{P_1V_1}{T_1} frac{P_2V_2}{T_2}) Volume and pressure are inversely related at constant temperature (Boyle's Law). Combined with temperature changes, the Ideal Gas Law provides a more comprehensive analysis.