91Ó°ÊÓ

Decrease the volume to one-fourth the original volume while holding the temperature constant. Since the temperature is constant, we can apply Boyle's Law, which states that \[P_1V_1 = P_2V_2\] where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume. To find the new pressure, divide the original volume by 4 and substitute into the formula: \[P_2 = \frac{P_1V_1}{(1/4)V_1}\] The \(V_1\) values cancel out, leaving: \[P_2 = 4P_1\] So when the volume is decreased to one-fourth of the original volume while holding the temperature constant, the pressure inside the cylinder increases by a factor of 4.

Reduce the Kelvin temperature to half its original value while holding the volume constant. Since the volume is constant, we can apply Gay-Lussac's Law, which states that \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\] where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature. To find the new pressure, divide the original temperature by 2 and substitute into the formula: \[P_2 = P_1 \frac{T_2}{T_1} = P_1 \frac{(1/2)T_1}{T_1}\] The \(T_1\) values cancel out, leaving: \[P_2 = \frac{1}{2}P_1\] So when the Kelvin temperature is reduced to half its original value while holding the volume constant, the gas pressure inside the cylinder decreases by a factor of 2.

Reduce the amount of gas to half while keeping the volume and temperature constant. Since both volume and temperature are constant, we can rearrange the Ideal Gas Law formula to solve for the pressure as follows: \[P = \frac{nRT}{V}\] To find the new pressure after reducing the amount of gas to half, substitute the new value for n and solve for P: \[P_2 = \frac{(1/2)nRT}{V}\] Comparing this to the initial pressure, \(P_1 = \frac{nRT}{V}\), we can determine the relationship between the new and initial pressure: \[P_2 = \frac{1}{2}\times (P_1)\] So when the amount of gas is reduced to half while keeping the volume and temperature constant, the gas pressure inside the cylinder decreases by a factor of 2.