Question

An ideal gas is contained in a cylinder with a volume of \(5.0 \times\) \(10^{2} \mathrm{~mL}\) at a temperature of \(30 .{ }^{\circ} \mathrm{C}\) and a pressure of 710 . torr. The gas is then compressed to a volume of \(25 \mathrm{~mL}\), and the temperature is raised to \(820 .{ }^{\circ} \mathrm{C}\). What is the new pressure of the gas?

Short answer

The new pressure of the gas after compression and heating is approximately \(5 \times 10^{4}\) torr.

Step-by-step solution

List the given quantities and convert temperature to Kelvin

Before we begin, it is important to list the given quantities and convert the temperatures to Kelvin since the Ideal Gas Law uses the Kelvin scale. To convert a Celsius temperature to Kelvin, add 273.15. Initial Volume: \(V_1 = 5.0 \times 10^2 \mathrm{~mL}\) Initial Temperature: \(T_1 = 30 .{ }^{\circ} \mathrm{C} = 303.15 \mathrm{K}\) Initial Pressure: \(P_1 = 710 \mathrm{~torr}\) Final Volume: \(V_2 = 25 \mathrm{~mL}\) Final Temperature: \(T_2 = 820 .{ }^{\circ} \mathrm{C} = 1093.15 \mathrm{K}\) We want to find the new pressure \(P_2\).

Use Ideal Gas Law for initial and final states

The Ideal Gas Law can be applied to both the initial and final states of the gas. We can construct two equations: Initial state: \(P_1V_1 = nRT_1\) Final state: \(P_2V_2 = nRT_2\)

Combine the two equations and solve for the final pressure

Since the amount of gas (n) and the Ideal Gas constant (R) do not change, we can equate the product of initial pressure and volume to the product of final pressure and volume: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) Now, we solve for the final pressure, \(P_2\): \(P_2 = \frac{P_1V_1T_2}{V_2T_1}\)

Substitute given values and calculate the final pressure

Now, we can substitute the given values and solve for \(P_2\): \(P_2 = \frac{(710 \mathrm{~torr})(5.0 \times 10^2 \mathrm{~mL})(1093.15 \mathrm{K})}{(25 \mathrm{~mL})(303.15 \mathrm{K})}\) \(P_2 \approx 5 \times 10^{4} \mathrm{~torr}\) Therefore, the new pressure of the gas after compression and heating is approximately \(5 \times 10^{4}\) torr.

Key concepts

Gas Compression

When a gas undergoes compression, its volume decreases while other properties, such as pressure and temperature, may change depending on the conditions of the process. In the case of an ideal gas, a hypothetical gas that perfectly follows the Ideal Gas Law without any intermolecular force influences or volume occupied by the gas molecules, these changes occur in a predictable way.

During compression, if temperature were kept constant (an isothermal process), the pressure of the ideal gas would increase due to Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume. In our problem, the compression is accompanied by a temperature increase, implying that the process is not isothermal. Both the decrease in volume and the increase in temperature act to increase the gas pressure. Therefore, to accurately determine the new pressure following compression, we need to consider both volume change and temperature change, as done through the application of the Combined Gas Law derived from the Ideal Gas Law.

Temperature Conversion

Temperature has a vital role in the behavior of gases and must be measured on an absolute scale for proper scientific calculations. The Kelvin scale is the temperature scale of choice in most scientific work because it is an absolute scale, starting at absolute zero, where all thermal motion ceases.

To convert from Celsius to Kelvin, we add 273.15 to the Celsius temperature. This is crucial when using the Ideal Gas Law, as it requires temperature to be in Kelvin. In our exercise, the given temperatures are first converted from Celsius to Kelvin. This avoids potential errors and ensures the accuracy of calculations involved in the Ideal Gas Law, which directly relates pressure, volume, and temperature of an ideal gas.

Pressure-Volume Relationship

The relationship between pressure and volume of a gas is central to understanding gas behavior under varying conditions. According to Boyle's Law, for a given mass of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. This means that when the volume decreases, the pressure increases proportionally, provided the temperature remains unchanged (isothermal process).

In the provided exercise, the initial and final pressures and volumes of the gas are related through the Ideal Gas Law, which effectively combines Boyle's Law with Charles's Law (the relationship between volume and temperature) and Gay-Lussac's Law (the relationship between pressure and temperature). The use of the Ideal Gas Law accommodates the changes in pressure, volume, and temperature simultaneously, allowing for a precise calculation of the new pressure after the gas has been compressed and the temperature has been raised.