Question

A sample of gas at \(p=1000 . \mathrm{Pa}, V=1.00 \mathrm{~L},\) and \(T=300 . \mathrm{K}\) is confined in a cylinder. a) Find the new pressure if the volume is reduced to half of the original volume at the same temperature. b) If the temperature is raised to \(400 . \mathrm{K}\) in the process of part (a), what is the new pressure? c) If the gas is then heated to \(600 . \mathrm{K}\) from the initial value and the pressure of the gas becomes \(3000 . \mathrm{Pa},\) what is the new volume?

Short answer

a) The volume is reduced to half. b) The volume is reduced to half, and the temperature is raised to 400 K. c) The pressure is increased to 3000 Pa, and the temperature is raised to 600 K. Answer: a) The new pressure when the volume is reduced to half is 2000 Pa. b) The new pressure when the volume is reduced to half, and the temperature is raised to 400 K is approximately 2666.67 Pa. c) The new volume when the pressure is increased to 3000 Pa, and the temperature is raised to 600 K is 2 L.

Step-by-step solution

a) Find the new pressure when the volume is reduced to half

In this case, the temperature remains constant, and the volume has been reduced to half. Using the Ideal Gas Law formula, we have: \(P_1 * V_1 = P_2 * V_2\) Since \(V_2 = 0.5 * V_1\), we can substitute and solve for the new pressure \(P_2\): \(P_1 * V_1 = P_2 * 0.5 * V_1\) Divide both sides by \(0.5 * V_1\): \(P_2 = 2 * P_1\) Now, all we need to do is substitute the initial pressure: \(P_2 = 2 * 1000 \,\text{Pa} = 2000 \,\text{Pa}\) The new pressure is 2000 Pa.

b) Find the new pressure when the temperature is raised to 400 K

Now, we need to find the new pressure when the temperature has been increased to 400 K while the volume remains half of the original volume. Using the Ideal Gas Law formula, we have: \(\frac{P_1 * V_1}{T_1} = \frac{P_2 * V_2}{T_2}\) Since we know that \(V_2 = 0.5 * V_1\), substitute the values and solve for \(P_2\): \(\frac{1000\,\text{Pa}\times1.00\,\text{L}}{300\,\text{K}} = \frac{P_2 \times 0.5\,\text{L}}{400\,\text{K}}\) Solve for \(P_2\): \(P_2 = \frac{1000\,\text{Pa}\times1.00\,\text{L}\times400\,\text{K}}{300\,\text{K}\times0.5\,\text{L}}\) \(P_2 = 2666.67\,\text{Pa}\) The new pressure when the temperature is raised to 400 K is approximately 2666.67 Pa.

c) Find the new volume when the temperature is raised to 600 K

In this case, we need to find the new volume when the temperature is increased to 600 K, and the pressure becomes 3000 Pa. Using the Ideal Gas Law formula, we have: \(\frac{P_1 * V_1}{T_1} = \frac{P_2 * V_2}{T_2}\) Substitute the given values and solve for \(V_2\): \(\frac{1000\,\text{Pa}\times1.00\,\text{L}}{300\,\text{K}} = \frac{3000\,\text{Pa}\times V_2}{600\,\text{K}}\) Now, solve for \(V_2\): \(V_2 = \frac{1000\,\text{Pa}\times1.00\,\text{L}\times600\,\text{K}}{300\,\text{K}\times3000\,\text{Pa}}\) \(V_2 = 2\,\text{L}\) The new volume when the temperature is raised to 600 K, and the pressure is 3000 Pa is 2 L.

Key concepts

Pressure-volume relationship

Understanding the pressure-volume relationship is crucial when studying gases. According to Boyle's Law, for a given mass of an ideal gas at constant temperature, the pressure of the gas is inversely proportional to its volume. This means that if you decrease the volume of the gas, its pressure will increase, provided the temperature remains unchanged.

For example, in our exercise, we observed that when the volume of the gas sample was reduced to half of its original volume, the pressure doubled. Mathematically, this principle is represented as:
\[ P_1 V_1 = P_2 V_2 \]
where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the new pressure and volume after the change. It's an important concept to understand because it demonstrates the inverse relationship between pressure and volume, a fundamental aspect of gas behavior under isothermal conditions (constant temperature).

Temperature-pressure relationship

The temperature-pressure relationship of gases is described by Gay-Lussac's Law, stating that the pressure of a gas is directly proportional to its temperature when the volume is kept constant. This is one part of the combined gas law, which is a combination of Boyle's, Charles's, and Gay-Lussac's laws.

Applying this to the exercise, when the temperature of the gas was increased from 300 K to 400 K, and the volume remained at half the original volume, the pressure increased. Using the Ideal Gas Law, we can show this relationship as:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
where the temperatures are in degrees Kelvin. By solving this equation, we find that the pressure adjusts proportionally to the change in temperature, which in this situation results in an increased pressure. This proportional relationship between temperature and pressure is key to understanding how gases will behave when heated or cooled in a closed system.

Gas law calculations

Gas law calculations are an indispensable part of understanding thermodynamics and the behavior of gases. The Ideal Gas Law provides a clear mathematical relationship between pressure (P), volume (V), temperature (T), and the number of moles of the gas (n). The equation is given by:
\[ PV = nRT \]
where R is the universal gas constant.

• The law allows us to predict the behavior of an ideal gas under different conditions of pressure, volume, and temperature.
• We can rearrange the equation to solve for any one of the variables if the other three are known.
• In our exercise, we used simplified versions of this law to find new pressures and volumes under changing conditions, as there are no changes in the number of moles and the universal gas constant remains the same.

By mastering these calculations, you can solve a variety of problems related to gas behavior and understand real-life applications, such as how a hot air balloon rises or how car tires respond to changes in temperature.

Thermodynamics

Thermodynamics is the study of heat and energy transfer, and it plays a critical role in understanding the Ideal Gas Law and real-life implications of gas behavior. The law itself is part of the larger field of thermodynamics, which examines relationships between various properties of gases, including pressure, volume, and temperature.

In our exercises, we've implicitly touched upon the First Law of Thermodynamics which is the principle of energy conservation. Any change in the internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings. When we compress a gas, we do work on it and increase its pressure; conversely, when we heat the gas, we're adding energy to the system, and thus its pressure or volume can increase if not constrained. These principles are fundamental to thermodynamics and are vividly demonstrated in the behavior of gases, which is effectively captured by the Ideal Gas Law in mathematical form.