Question

(a) One liter of gas with \(\gamma=1.32\) is at \(273 \mathrm{~K}\) and \(1.00 \mathrm{~atm}\) pressure. It is suddenly (adiabatically) compressed to half its original volume. Find its final pressure and temperature. \((b)\) The gas is now cooled back to \(273 \mathrm{~K}\) at constant pressure. Find the final volume. \((c)\) Find the total work done on the gas.

Short answer

Finally, the final pressure after the adiabatic process is approximately 1.874 atm, the final temperature is around 364K, the final volume after cooling is approximately 0.375 liters and the total work done on the gas is about 3.99 J.

Step-by-step solution

Calculation of Final Pressure and Temperature after Adiabatic Compression

For an adiabatic process, \(P V^\gamma = \text{constant}\) and \(T V^{\gamma-1} = \text{constant}\). So, by applying these formulas, the final pressure \(P'\) and final temperature \(T'\) can be found. Using the given data and these equations, \(P' = P (V/V')^\gamma = 1.00 \text{ atm} \times (1/0.5)^{1.32} \approx 1.874 \text{ atm}\) and \(T' = T (V/V')^{\gamma-1} = 273K \times (1/0.5)^{1.32-1} \approx 364K\).

Calculation of Final Volume after Cooling at Constant Pressure

As the gas is cooled back to its initial temperature under constant pressure, we can use the relation \(V/T = \text{constant}\) for isobaric process to find the final volume \(V''\). So, \(V'' = V' \times T/T' = 0.5 \times 273/364 \approx 0.375 \text{ liters}\).

Calculation of Total Work Done on the Gas

The work done for an adiabatic process can be calculated using the equation \(W = \frac{P'V' - PV}{\gamma - 1}\). Inserting the given and calculated values for \(P', V', P\), and \(V\) we get \(W = \frac{1.874\text{ atm}\times0.5\text{ L} - 1\text{ atm}\times1\text{ L}}{1.32 - 1} \approx 3.99 \text{ J}\). For the second part of the process (cooling), no work is done as it is under constant pressure and the volume is decreasing (work done by the gas is zero). So, the total work done is 3.99J.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It is governed by four fundamental laws, which lay the framework for various processes that occur in nature. An adiabatic process, such as the one described in the exercise, is a thermodynamic process where no heat is transferred into or out of the system. Consequently, all the work done on the gas results in changes to the internal energy of the system, thus affecting its pressure and temperature.

Understanding thermodynamics is essential for physics problem solving, particularly within the scope of processes involving gases. This field provides valuable insights into how energy transformations underpin a wide range of natural and industrial activities.

Isobaric Process

An isobaric process is characterized by constant pressure throughout the entire system. Any changes in volume or temperature occur without changes to pressure. In the given exercise, the cooling of the gas back to its original temperature is a typical example of an isobaric process. When we apply the concept of an isobaric process to a physics problem, we use Charles's Law, which states that at constant pressure, the volume of a gas is directly proportional to its temperature (in Kelvin).

In practical terms, if you cool a gas and the pressure remains unchanged, its volume must decrease; conversely, heating the gas up would increase its volume. Knowing this law facilitates the calculation of the final volume of the gas after it is cooled, which is an integral part of solving problems in thermodynamics.

Physics Problem Solving

Solving physics problems involves a systematic approach that combines conceptual understanding with mathematical skills. The problem given is a great example of this. It requires students to apply their knowledge of thermodynamics and gas laws to find solutions. To tackle such problems, one must:

• Identify which physical laws apply to the scenario.
• Apply relevant equations representing these laws.
• Perform correct mathematical operations to find the desired quantities.

Improving one's skill in physics problem solving frequently involves working through a diversity of problems, which can help in understanding the different ways physical laws interact with one another in various scenarios.

Gas Laws

Gas laws are the mathematical relationship between the pressure, volume, temperature, and amount of a gas that describe how they affect each other. For instance, Boyle's Law would apply to an adiabatic process involving pressure and volume, while Charles's Law relates to an isobaric process involving volume and temperature. To find the final pressure (\(P'\)) and temperature (\(T'\)) during an adiabatic compression, we use the adiabatic laws, which are a specific subset of gas laws.

By mastering gas laws, students can predict how changes in one state variable affect others, making it a crucial part of thermodynamics and physics problem solving. They enable a structured approach to assessing physical situations, such as determining the final state of a gas after compression or expansion.