Work Done by a Gas Examples: Isothermal and Adiabatic Processes

📚 Quick Study Guide

• 🌡️ Isothermal Process: A process occurring at constant temperature ($T$). Work done is calculated as: $W = nRT \ln(\frac{V_2}{V_1})$, where $n$ is the number of moles, $R$ is the gas constant, $V_1$ is the initial volume, and $V_2$ is the final volume.
• 🔥 Adiabatic Process: A process where no heat is exchanged with the surroundings ($Q = 0$). Work done is calculated as: $W = \frac{P_2V_2 - P_1V_1}{1 - \gamma}$, where $P$ is pressure, $V$ is volume, and $\gamma$ is the heat capacity ratio ($C_p/C_v$).
• ⚙️ Key Differences: Isothermal processes maintain constant temperature, while adiabatic processes involve temperature changes.
• 💡 Important Note: Remember the ideal gas law, $PV = nRT$, which is often useful in solving these problems.
• 🧮 Sign Convention: Work done *by* the gas is positive, and work done *on* the gas is negative.

Practice Quiz

1. Which of the following is true for an isothermal process?
   1. A) Temperature remains constant.
   2. B) Heat remains constant.
   3. C) Pressure remains constant.
   4. D) Volume remains constant.
2. What is the work done in an adiabatic process if $P_1 = 2 \text{ atm}$, $V_1 = 1 \text{ L}$, $P_2 = 1 \text{ atm}$, $V_2 = 3 \text{ L}$, and $\gamma = 1.5$?
   1. A) -2 L atm
   2. B) 2 L atm
   3. C) -4 L atm
   4. D) 4 L atm
3. In an isothermal expansion, what happens to the internal energy of an ideal gas?
   1. A) Increases
   2. B) Decreases
   3. C) Remains constant
   4. D) First increases, then decreases
4. Which of the following processes involves no heat exchange with the surroundings?
   1. A) Isobaric
   2. B) Isochoric
   3. C) Isothermal
   4. D) Adiabatic
5. A gas expands isothermally and its volume doubles. If the initial pressure was 2 atm, what is the final pressure?
   1. A) 4 atm
   2. B) 1 atm
   3. C) 2 atm
   4. D) 0.5 atm
6. For an adiabatic process, which of the following relationships is correct?
   1. A) $PV = \text{constant}$
   2. B) $PV^\gamma = \text{constant}$
   3. C) $P/V = \text{constant}$
   4. D) $P/V^\gamma = \text{constant}$
7. What is the value of work done if volume remains constant?
   1. A) Maximum
   2. B) Minimum
   3. C) Zero
   4. D) Cannot be determined