A fixed quantity of gas at 25 _x001F_C exhibits a pressure of 99 kPa and occupies a volume of 4.00 L. (a) Calculate the volume the gas will occupy if the pressure is increased to 202.6 kPa while the temperature is held constant. (b) Calculate the volume the gas will occupy if the temperature is increased to 100 °C while the pressure is held constant.

Verified step by step guidance

1. Step 1: Identify the initial conditions for both parts of the problem. For part (a), the initial pressure \( P_1 \) is 99 kPa, and the initial volume \( V_1 \) is 4.00 L. For part (b), the initial temperature \( T_1 \) is 25 °C, which needs to be converted to Kelvin.
2. Step 2: For part (a), use Boyle's Law, which states that \( P_1 V_1 = P_2 V_2 \) when temperature is constant. Rearrange the equation to solve for the final volume \( V_2 \): \( V_2 = \frac{P_1 V_1}{P_2} \). Substitute the known values: \( P_1 = 99 \) kPa, \( V_1 = 4.00 \) L, and \( P_2 = 202.6 \) kPa.
3. Step 3: For part (b), convert the initial temperature from Celsius to Kelvin by adding 273.15: \( T_1 = 25 + 273.15 \). The final temperature \( T_2 \) is 100 °C, which also needs to be converted to Kelvin: \( T_2 = 100 + 273.15 \).
4. Step 4: Use Charles's Law for part (b), which states that \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) when pressure is constant. Rearrange the equation to solve for the final volume \( V_2 \): \( V_2 = V_1 \times \frac{T_2}{T_1} \). Substitute the known values: \( V_1 = 4.00 \) L, \( T_1 \) and \( T_2 \) in Kelvin.
5. Step 5: Calculate the final volumes for both parts using the rearranged equations and substituted values.

Key Concepts

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. This means that if the pressure increases, the volume decreases, and vice versa, as long as the temperature remains unchanged. Mathematically, it can be expressed as P1V1 = P2V2, where P is pressure and V is volume.

Charles's Law

Charles's Law describes how gases tend to expand when heated. It states that the volume of a gas is directly proportional to its absolute temperature when pressure is held constant. This relationship can be expressed as V1/T1 = V2/T2, where V is volume and T is temperature in Kelvin. Thus, increasing the temperature of a gas will result in an increase in its volume.

Ideal Gas Law

The Ideal Gas Law combines Boyle's Law, Charles's Law, and Avogadro's Law into a single equation: PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. This law provides a comprehensive framework for understanding the behavior of gases under various conditions, allowing for calculations involving changes in pressure, volume, and temperature.